I've had something very similar a couple of times in the last couple of years too, since my wisdom tooth came out... the ones on the left side of my mouth are both fine, no problems at all, in fact I didn't even notice they'd come out... but my bottom right one hasn't come out and the one above it sticks out at a funny angle.
What's happened to me a couple of times is that the area where the bottom tooth should be has either swelled up for some reason or the tooth suddenly decided to try to push through, I'm not sure which, but either way I couldn't close my mouth properly because of the upper tooth pressing on it. I didn't go to a dentist either time though, it just went away again by itself within a day or two, so maybe it was something different and not as severe that caused it for me.
Right now I am having a different problem though, which is how I've ended up here (looking for any information on the symptoms I've currently got, or anyone with the same symptoms, but not found anything at all that sounds similar)... I've had a cold for the last week or so, and since a couple of days ago the area around the upper right wisdom tooth that's at a funny angle feels odd... it kinda throbs and hurts if I step too heavily when walking, or shake something in my hand too vigorously etc... but even just going from lying down to sitting up, and sitting to standing up etc seems to cause it too, even if done very slowly and smoothly... and the teeth themselves don't hurt when touching them etc, it's kinda right up in the gums and in the side of my face (although there's no visible swelling or anything)... so I'm wondering if I've got some kind of secondary infection from the cold that's got in there somehow, and maybe I need some antibiotics to get rid of it... although I guess that tooth probably needs to be removed anyway because of the way it's sticking out oddly and pressing into where the one underneath should be :-\
I'm surprised the dentist doesn't want to remove yours though... I asked my dentist a few years ago if they could do anything about the one that had come out at an odd angle, and they said they couldn't do anything to correct it but they could remove it if I wanted them to, but that there wasn't really any need to if it wasn't causing any problems... so I'm kinda surprised that yours doesn't want to remove one that clearly is causing problems for you and that you actually want removed.
To Jessie, the person who replied before me... it sounds like your teeth were in a very similar situation to mine (or two of mine anyway), and you said you didn't feel a thing when you had them taken out... so what kind of anesthetic did they use?