No, you shouldn't just give up, but you do need to take steps to try and unravel this mess.
Gather all the documentation you have about the amounts being taken from your check for support. You need to contact the child support division and try and straighten this out. It may take a fraud complaint through the police department or the DA's office, but you should be able to get this fixed.
A driver's license should only be suspended for a willful failure to pay support. This one sounds like it was out of your control. They should be able to recognize this, provided you can prove all this to their satisfaction.