As a Vimes fan, I can't deny that it was satisfying to have him so definitively be the tarnished sun around which the book revolves. Still, I really don't read Carrot leaving as an attempt to reassert his slipping dominion/ownership over Angua at all. To me, it seems like a fairly understandable move for anyone in a relationship. Angua leaves, without notice to anyone, following a period of some emotional turmoil, likely for somewhere that Carrot knows is a location of some unrest given the Low King situation, and he's supposed to- what? Sit there like a good Carrot and pretend he doesn't have a girlfriend? He's already demonstrated worry about her absence to Vimes & co. when they departed for Uberwald. Angua's been building up to this for some time, starting in FOC, and her leaving and Carrot following is, in a way, foreshadowed in Jingo. Carrot cares for Angua, and he will follow and make sure of her safety- it's really very in character.
It's been mentioned before that Carrot believes that the personal is not the same as important, but in this case the two very much overlap for him. That he turns in his notice reflects how much he understands that. Like raisindot mentioned, I think this shows that he would be willing to settle down in Uberwald with Angua, if that's what she wants. He understands that Angua is distressed over the "werewolf thing" and about their future together (and has been for a while), and this reads to me like a fairly big indicator to Angua of just how important she is to him. It's also growth in an interesting direction for typically uber-responsible Carrot- him placing his and Angua's relationship over his work shows him breaking away from the kingly, perfect Carrot. Yeah, it's selfish, but that's human and lookie here- Carrot's finally, finally human. Carrot's still himself in TFE, but here the focus of his storyline is very much on the personal scale rather than the usual dramatic scope his actions usually dwell in. In TFE, that's in Vimes's wheelhouse. I don't look askance at Carrot's storyline being so concentrated on the personal or dismantling Carrot's image for something more three-dimensional- for me, it's an obvious step because so far we've only seen one Carrot and he's far too gleaming to exist in DW.
Wrt Gavin: It's not like he deliberately creates a third-wheel- if anyone is responsible for that, it's Angua, who honestly seems like the most selfish/irresponsible player here (which is not to say that I'm not terribly fond of her and understand her struggle). She could have left a note, she could have confronted Carrot but instead she just slips off, knowing full well that Carrot would follow.