“There are two aspects you need to consider. First, is the dealership asking you to pay them for the $500 sales or excise (it’s one or the other) tax,”
In some states it could be Personal Property Tax.
" or are they advising you that $500 is still due?"
They let someone out the door with a car with $500 still due? Hmmm.
“Read the letter carefully, since non-payment of sales tax (yes, even after the purchase and registration is complete) is your problem, not the dealership’s.”
In most jurisdictions the merchant is responsible for payment of sales tax whether or not it was collected from the customer. When dealing with cars “sales tax” is really a “you bought a car tax” and may be handled differently.
“Your RMV or DMV or what have you WILL suspend your car’s registration”
In which case they will most likely send you a letter.
“and nullify your insurance”
Whoa, there, hoss! The State can’t nullify a contract between you and your insurance company.
“and may even suspend your license to drive.”
Seems unlikely. On what grounds?
“If you get pulled over on the road and these things have happened, you will pay a lot more than $500. You may even need a lawyer. You can sue the dealership if they are at fault. But you have a verbal agreement, not a written contract. You’d need to prove your position in a civil court. You have the burden of proof.”
The proof is that the dealer let the OP have the car. The burden is on the dealer.