I also don't like vomiting (my fear of flying isn't about the flying itself, but about the fasten seat belt being on and needing to go to the bathroom to vomit if there were super bad turbulence

). Even when I just have sedation from a procedure like a colonoscopy I feel a little sick to my stomach after. But like others have said at least your system is totally "empty" so if you do get sick you aren't actually making any mess. Also, after my last colonosocopy a couple weeks ago I told them before that I do feel sick after sometimes and they did give me a drug which did help me. It didn't make the upset feeling go away entirely, but it helped a lot.
The thing is, that I realized it's not really too bad. Although I might wake up from my upcoming surgery sick to my stomach the feeling does go away after a bit, and the drugs do help. So I definitely share your anxiety about it, but I think after going through another procedure a couple weeks ago and getting sedated I guess it has helped calm my nerves a bit because I know the feeling isn't too bad, I won't make a big mess even if I do get sick, and the feeling goes away after a little bit. So don't be too worried.