Does cutting your hair regrow it into finer hair?

Depends what you want.  Cutting it won't regrow it finer, if anything it will feel thicker because it will all be blunt cut into the same length. 

If you want to cut it and regrow it just to do it, then that's what you should do.  If you only want to cut it because you like it longer and think it will be better the next time you grow it, you probably should try to figure out what made it dry and see if you can fix it first.