Do not grind down your teeth to correct your bite - this is horrible treatment and can lead to TMJ problems in most situations. It is a philosophy from the 1960s that should be abolished. If your teeth are out of alignment significantly, then they musth be treated first with...
If what you say about your treatment is as described, then I think it is unusual for your doctor to charge these additional fees. Minor corrections after brace removal are not uncommon, so doctors I am aware of would not charge additional for a patient who followed their instructions...
Your bottom teeth will never be straight on their own. If anything the overlapping will get worse as you get older.
This can be treated with traditional braces or Invisalign.
I would make an appointment with an orthodontist or a certified Invisalign provider for a consultation.