So this is my take on if Han was Force sensitive. I'm taking a little liberty with the story line, but not much.
Disclaimer: I don't own them, I just enjoy creating situations for 'em.
Hearing the old man talk to the kid about the Force made Han feel very uncomfortable. Han had claimed not to believe in it, but that was easier for him to convince others, and more importantly himself, than the real truth.
The Force had brought nothing but pain to Han for as long as he could remember. After all it was because of the Force that Han's mother had been killed.
Han had been a young boy when the Empire first started to take over the galaxy; trying to eliminate every force sensitive being they could find. Han had gotten out alive, but his mother, Jaina, hadn't been so lucky. Every once in a while she would pop up as a Force ghost, but most of the time she just communicated with him in his thoughts.
After that Han really only used the Force when he felt it was necessary, like winning the Millennium Falcon from Lando. The older he got to be, the less he wanted to have this ability, and wished it would just go away. Little did he know, his powers were going to be incredibly helpful.
After all the training with the lightsaber and remote, Luke was tired and had fallen asleep on the acceleration bench and using the holochess table as a pillow. Chewbacca was preparing a snack for the group, so it was Han and Obi-Wan in the cockpit. They had been sitting in silence when Obi-Wan spoke.
"The Force flows within you, much like it flows within the boy" Han tensed up, but tried not to let old man see any reaction. "You say you don't believe, but I think you just find it easier to try and ignore. I must warn you, that is not a path you have a choice of taking."
"Oh yeah," Han fired back, "it seems to be working so far." He looked at of the panels that he had just been looking at, trying to appear busy, but it wasn't working.
"You try to pretend you don't care about anybody but yourself, but I can tell you care very deeply and it scares you. You need to let go of the fear. Fear leads to the Dark Side. I have seen it first hand Han, and its a devastating thing. Don't let that happen to you. You have been destined to do many great things if you let the Force help."
Han scoffed. "You can't be suggesting I do those stupid training exercises that Luke is doing."
"No no, those won't do you any good. Those are for Jedi, and you are not meant to be a Jedi. You have no use for a lightsaber. I do suggest meditation and exploration of your powers in the Force, then you become a better Force user. It will be of great benefit to you. I can teach you how to do that if you would like."
A person didn't need the Force to sense Han's hesitancy.
"It will stay between you and I. I swear." Obi-Wan assured him.
Jaina had taught Han some mediation and exploration tricks when he was young, but not nearly as much as she would have been able to had she lived. He knew how to find other Force sensitive people, and how to sense situations around him which was good in the smuggling business, and usually when he used the Force.
A bit reluctantly, Han took the old man up on his offer. Before they could get started though, Chewie came in and told them he had just made a meal instead and that it was ready.
They woke Luke up to eat, and before Han could finish, the proximity alarm sounded. Both he and Chewie rushed to the cockpit to make the landing only to find themselves rapidly approaching an asteroid field.
They navigated with special maneuvers, and a little bit of the Force, through the asteroids and approached what appeared to be an uncharted moon.
As it turned out, that "moon" was actually and Imperial space station, and they found the princess Han didn't even know they were looking for. Han, Luke and Leia, as Han found out the princess' name was, had a rough escape, but no one had died before they reached the dock where the Falcon was. Just as they were running towards the ship, they spotted a lightsaber battle between Obi-Wan and Darth Vader. All three of them watched as Obi-Wan was struck down by Vader and just disappeared. Luke cried out and Han felt for the kid, but they needed to get on the Falcon and leave. They finally made it out of the space station and after a minor altercation with some tie fighters, were traveling comfortably through hyperspace.
Once things had settled down and they were on course to the rebel base on Yavin IV, Han took a moment to feel out in the Force. He felt two Force sensitive beings other than himself, The only possibilities were Luke and Leia. He already knew Luke had the Force, but Leia didn't give off any inclination of knowing she was Force sensitive. Han didn't really give a second thought to it though because it wasn't like he went around announcing his Force sensitivity. Their Force signatures were very similar but there was something quite distinct that set them apart.
Deciding the Falcon was fine in autopilot, he walked out to the lounge where the others were. Her Royal Highness and the kid were sitting at the holochess table quietly talking while Chewie was fixing something to eat. The general atmosphere felt melancholy and its was almost choking Han.
"Hey kid, I'm sorry about the old man." Han said to Luke as he sat down beside the younger man.
"He had a name you know." Luke glared at Han. Leia just sat there rubbing his Luke's back, almost mechanically.
Han saw the glazed over look in her eyes and immediately knew she was having a flashback to something. Han made a mental note to talk to her alone later. He also knew better than to sneak up on her or touch her.
He could only imagine what the Empire had done to her. He had learned the basics of their torture methods while he was in the Academy. It was required, and it made him sick to his stomach. Han hated hurting people unless he had no other options. The only exception was if someone threatened a person he loved.
Chewie walked out of the kitchenette and placed a plate in front of Luke first and then Leia. Han saw Leia jump and panic momentarily, not aware of her surroundings. He reached out to her through the Force the best he could. He watched her calm down and find her bearings. They ate their snacks. Well more like Han and Chewie ate, Luke and Leia just pushed it around.
"I'm sorry Chewbacca, it's not your food, I'm just not hungry." Leia apologized.
Things were feeling awkward around the table, so Han spoke up.
"it's going to be another 12 hours before we reach the base. I'm going to take the first shift in the cockpit while Chewie catches a few hours of shut eye. You two are more than welcome to get some sleep as well. Luke you know where everything is at. Leia, let me show you around quick." Han offered, smirking at her and holding out his hand for her to take. Leia just stared at him like he was completely insane.
"I'm pretty sure I can find everything quite alright Captain." she replied coldly.
Han didn't appreciate the attitude she was giving him. He was just trying to be kind after everything she had likely been through.
"Well fine Your Worship. But don't come asking me for anything she you can't find it." Han smarted back. He huffed his way to the cockpit, away from snooty royals.
Han heard the door open a few minutes after he stormed off.
"Go away!" he demanded of whoever entered.
"I'm one of the last people you wanna be giving attitude, cub." Chewie howled.
"Sorry bud, thought you were one of the wonder twins." Han said rolling his eyes.
"I just wanted to ask where you wanted to put the princess. Luke has the extra bunk already." the wookiee informed him.
"She can sleep in the smuggling compartments for all I care." Han grumbled.
"Hey." he barked, "You know she didn't mean anything by what she said. She's been through a lot and you're still practically a stranger to her.".
"Yeah well so is Luke and she didn't have an issue warming up to him." he complained.
"Oh… so that's what this is about. You're jealous of the kid. Well you two started this entire thing off wrong. You've been screaming and snarking since the beginning. Why don't you try talking to her like a regular human being, and see how that works." Chewie suggested.
Han sighed. Chewie had a point. The two of them had gotten off on the wrong foot.
"Besides she is a lady and you will treat her well." Han could hear the warning that Chewie was giving him.
"I would never do anything to her. You know I'm not that kinda guy." Han pointed, and then took a breath, deflating. "And I guess you're right about being nice. I will do it later. I can sense she's still mad at me. Just uh, give her my cabin. I'll just climb into your bunk when we switch. Make sure you grab my spare blanket though. I don't wanna be covered in wookiee hair because I used yours." All Han heard was a growl of confirmation that he had been heard as Chewie walked away.
Unfortunately for Han, Leia spent all of her time avoiding him. By the time they reached Yavin, he was completely past the point of civility. If she wasn't going to play nice, then neither was he.
He was getting his reward loaded onto the Falcon and had tried convincing Luke to leave with him, but Luke refused.
The Falcon left before the X-wings could take off, and Han was about to punch it into hyperspace when he got the sinking feeling he needed to go back and fight. He tried to ignore it when his mother's voice popped into his head.
"Han, you need to go back. They need you." she whispered to him.
"I need to get back and pay off Jabba." he replied without talking.
"Jabba will be there later. Leia and Luke might not be. We both know you care about them, no matter how much you try to say otherwise. They are still young and could use you. Not to mention everything Leia has lost and been through. Both of them really. You know what it's like to lose everything. Imagine if you had someone around who knew what you were going through. Now turn back around and go save them." And she was gone.
Han couldn't deny what his mother was telling him. He went out to the hall where Chewie was rearranging tools and told him they were turning back around. Chewie gave an excited howl and told Han he would begin preparing the weapons.