Hey Madi,
I'm so sorry to hear about your experience with that dentist! That sounds absolutely horrible. I've also had problems in the past with numbing agents, and my dentist would always tell me to raise my hand (but not the one closest to her, because then I might punch her out or something) if something was bothering me so that she could correct it. I remember that there was one time in particular when she gave me more of the numbing agent because I was experiencing pain. If you are feeling pain, definitely tell them! That should not happen under any circumstances. I don't know if your parents would be open to this, but it might be worth asking them if you can switch dentists because of your past experience. However, because they don't believe that anxiety is real, they might not be open to this idea, unfortunately.
Teen years can be tough because you don't always have the ability to pick and choose where you go or what doctors you see.
If you are noticing cavities, I highly recommend getting them filled now. You're young, and if you let them go too long, it's possible that you might need a crown or a root canal, which can be pretty expensive. A root canal in particular is a fairly invasive procedure, so it's always best to avoid that. But I know that this is hard because of your last experience with the dentist - after that, it's very understandable that you would have anxiety about going! My advice, if you're not able to change dentists, is to communicate with them to the best of your ability. Tell them up front that last time the numbing agent wasn't effective and that you were in pain, and also ask if they can give you a warning prior to giving you a needle or sticking things in your mouth. I hope that this is something that they would be receptive to as professionals.
Although it's scary, it's really important that you get those cavities taken care of sooner rather than later.
Remember that doing the fillings now is ultimately better than having to go in for a root canal, which generally takes longer, is more invasive, and is more uncomfortable. If you're feeling anxious during the fillings, try to practice visualization; that's something that's helped me in the past with these kinds of procedures. Pretend you're doing something that you love, such as a hobby or spending time with an animal. Try your best to keep your mind off the procedure itself, and remind yourself that it will be over soon. That's something that also really helps me - reminding myself that the end is in sight. When I was little, my mom used to tell me, "The sooner you do it, the sooner it will be over," and I've found that it's really true!
I really hope this helps.
I wish you the best of luck!