The problem with Carrot is an intriguing one--and one which begins to become clear in Feet of Clay. Pratchett made him a naive, dwarf- indoctrinated figure in his first appearance (G!G!) for a number of understandable reasons. First and probably foremost, he is a marvellously funny character. He arrests the head of the Thieves Guild for being a thief. He learns and tries to enforce all the statutes in the codified (and clearly out of date legally) Ankh--Morpork code of laws. But Terry also began the suggestion that he is somehow the heir to the throne (and therefore "divinely right) in this book. Unfortunately those two elements make it increasingly impossible to believe in Carrot as a person. It also makes it impossible for him to learn and develop, and this eventually makes him so boring that increasingly, Terry shunts him into a minor roles.
Carrot is terminally naive, and his moral code is essentially that of a dwarf. Since coming to AM, he has substituted Sam Vimes as his father figure, and tends to take Sam's orders as gospel--if he thinks about them at all. But, he follows orders from Vimes only when it suits him. (eg. - goes into the candle factory when he has been specifically told not to do so).
And for all his "interest" in other people and species, he never really understands them because, I think, he sees them through dwarf-colored blinders. He is horrified at Cheery's revelation that she is a female dwarf. He fails to understand that Angua is different -- a werewolf. In fact, he has no real understanding of other species or humans.
And thus, Terry lets us know in the last lines of the novel that Carrot is unable to change. After all they have been through together, he assumes that Angua (though he knows she is a werewolf) sees the world as he does. He expects her to understand that he has to write his letter home first (of course), and then he's come to ask her not if she will marry him, but if she'd like to help him fix up the Dwarf Bread Museum!