I had my bottom wisdom tooth taken out with only novaicane and it wasn't pleasant, but mine was impacted in the bone. I am not sure exactly how it works but if your tooth is under the gum but not impacted in the bone, you will probably be fine with local. Because I have had gum surgery with just novaicane and I was fine. But if it is in the bone, those are quite difficult to get out, you should ask your dentist why he doesn't think that you need general.
But for my top teeth they were slightly out and the dentist gave me novaicane and pulled them out in three seconds. It didn't hurt, I didn't swell afterwards and it was quite uneventful.
You should talk to the dentist again about your concerns and maybe ask to see your xrays, so that you can see for yourself whether your teeth are impacted or not. Best of luck to you