Can they repossess your car even if you offer to pay them your late payments then sell it and expect you to pay the difference?

In most states and under most conditions YES. Read your contract.

You broke the contract. It is their choice as to how to handle it, not yours. If you sign an agreement, you have to honor it the way it is stated. Often it is not up to you as to when to pay or how much and they are not your mom.