First, welcome! And, yes, you are in the right place.
Extractions are actually generally way quicker and way easier than fillings. My top 2 wisdoms teeth (the ones that had already come in) were both taken out in less than five minutes. I didn't even realize they were because I didn't feel it at all! It took longer to get numb than to have the teeth extracted! After, you may have some discomfort, but nothing that over-the-counter pain medication won't fix.
Because you are younger than I was, it should be even easier because the roots may not be as fully formed yet.
I also went periods of many years without going to a dentist. I even went at one point when I was about your age (hadn't been since I got my braces off when I was 17). I saw a dentist a few times and then she got sick and I didn't go back again until I was 30yo. When I went back then (at 30), I finally found an awesome dentist who I've been going to for more than 7 yrs now. It's weird to say, but it's just in the last 6 months or so that I've stopped being horribly anxious every time I go. So, I'd encourage you, once you're out of this emergency situation to try to find a dentist who you can start to trust so that you can begin to have regular dental care - it will save you a whole lot of troubles in the long run!
Best of luck to you and please let us know how it goes.