The following is my interpretation on Shagrat's fate in Lord of the Rings and the Return of the King.
Shagrat raced through Cirith Ungol with Frodo's mithril coat. He was able to avoid running into Sam, who killed several orcs in the tower, including Gorbag.
Shagrat had wanted to take the mithril coat for himself, while Gorbag had insisted that it be taken to Sauron. This resulted in a fight, in which Shagrat had told the other orcs to kill Gorbag. When Gorbag kicked an orc out the window, he landed on other orcs, causing them to spill their beer, resulting in a civil war between the orcs, of which he was the only survivor.
Shagrat knew Sauron would be angry with him for the chaos he caused in Cirith Ungol. He was the reason all the orcs in the tower were dead, the only defense against an attack from the mountains if one were to get by Shelob. He could only hope that Sauron would forgive him if he brought him the mithril coat himself.
Shagrat made his way into Mordor. He ran as fast as he could to Barad-dûr, and climbed up to the top, so that he stood next to the Eye of Sauron.
"Master!" said Shagrat. "I found a Halfling. He didn't have the ring on him, but he did have this." He held up the mithril coat and placed it next to Sauron.
"Where is the Halfling now?" said Sauron.
"Well," said Shagrat, "Gorbag wanted to take that shiny shirt for himself, and because of him, our men killed each other."
"I see your greed," said Sauron. "I know who is truly responsible."
Sauron shot a mighty blast of flame at Shagrat, incinerating him in a matter of seconds.
Moments later, he became aware that an army led by Gandalf and Aragorn was on its way to the Black Gate, and so he called the Mouth of Sauron to take the mithril coat to the gate.
AN: If you haven't seen the extended edition, the Mouth of Sauron is a servant of Sauron talks to them at the gate before the army of orcs emerges, and has his head cut off by Aragorn.
Also, I edited the attack he used to something I thought would be more realistic, besides, we've seen his darkness engulf Thrain when he was a cloud of darkness, so it makes sense he'd attack with fire as a flaming eye.