Hmm. That site does seem to suggest that Shagrat was killed by Sauron... but..
(The following are the sources given for that website's section on Shagrat.)
The Two Towers: "The Choices of Master Samwise," p. 344-52
The Return of the King: "The Black Gate Opens," p. 165-66; "The Tower of Cirith Ungol," p. 174-84, 187
Appendix B of The Lord of the Rings: "The Tale of Years," p. 375
The Lord of the Rings: A Reader's Companion by Wayne G. Hammond and Christina Scull: "The Land of Shadow," p. 608
but... I went back and read the first three of the sources and not one of them mentions the death of Shagrat. The last words regarding Shagrat in the RotK are:
"Away below Shagrat went leaping down the stairs and out over the court and through the gate, bearing his precious burden. If Sam could have seen him and known the grief that his escape would bring, he might have quailed."
Shagrat is last mentioned in Appendix B under March 17, 3019 where it states that he brings the cloak, mail, and sword to Barad-dur. Nothing about his death.
The text in TT is irrelavent.
So unless it states in "The Lord of the Rings: A Reader's Companion" that Shagrat was killed by Sauron, we have no reason to suspect such a thing. Unfortunately, I don't have a copy of that book to check.
I also checked EoA and Wikipedia and neither of them mention Shagrat's death...
So for now... I'm still unconvinced.