I think it's the mummy bit of you - since you've had kids - it's natures way of making you over-think things. anything seen as an immediate threat to you (and consequently your children) can become overblown, and seen through a mummy prism, to protect your children.
rationally, you know you will be fine. emotionally, somewhere deep down, you are thinking 'arrghhh!'
Like when ds1 (he was 11 at the time) needed a general anaesthetic for plastic surgery on his finger - I was very cheery, took him into the anaesthetic room, joked and stroked him whilst he was going under - then the tears ran down my face. The theatre staff were brill, and I walked back to the ward sobbing to the ward nurse. I knew rationally that he would be absolutely fine. I absolutely knew it. I am a nurse FGS! didn't stop me crying though.
I've also had general anaesthetics myself since children, and each time, I've felt more uneasy than what I did pre-children - rationally, I know things will be fine... But I did feel slightly more anxious...
BTW- I've been a nurse for 24 years, and have never come across a 'bad' reaction to an anaesthetic.
I think what you have to tell yourself is that 'This is very very safe - I am going to be fine. I feel anxious, because I will feel out of control and I have the kids to think about - It's normal to feel anxious when I have kids - it's natures way of making sure I don't parachute without a parachute, hence the anxiety. But am not going parachuting - am having a very safe procedure under a very safe anaesthetic, so instead, am going to focus on feeling well.."
You will be fine!! honestly!! but when you go in - do let the staff know you are feeling anxious - a friendly face, and some emotional support goes a long way...