Ugh OP, I was absolutely TERRIFIED of my first adult GA - I'm severely emetophobic and hadn't had one since I was 12, and whilst I wasn't sick then I did feel it.
I'd hoped to get through adult life without ever needing another, but alas my gallbladder put paid to that when I was 45. I spent weeks dreading it and the poor anaesthetist on the day had to listen to my frantic insistance on being given nothing to make me nauseous - no anaesthetic gases, no morphine (I demanded iv only anaesthetic and paracetamol and said I didn't care about the pain), and maximum anti-emetics.
He was very kind and reassuring, and although I was frightened of losing control, actually going under is quite pleasant. Your eyes go a bit swimmy, then it's just lights out. No idea if I had a breathing tube for that but even if you do, you won't know anything about it other than a slightly scratchy throat afterwards.
When I woke after that surgery I did feel a bit iffy - but they been pulling everything around in my abdomen and they also gave me fentanyl as an alternative to morphine. Turns out I don't do so well on fentanyl, I felt very nauseated and couldn't move my limbs for 9 hours, everyone else was heading home and I was still flat out in bed!
However - I've had an orthopaedic surgery since then, under GA, and I told the anaesthetist the same as before but that I took hours to recover from fentanyl. They gave me alfentanyl instead, and I was right as rain - woke up after a lovely deep refreshing snooze and was trying out my crutches within 30 minutes of waking.
I've also had double endoscopies under propofol sedation, nothing to it. I've feared GA all my life and now I'm not bothered!
Good luck, you'll be absolutely fine.