@sarahprice,
Even if your fiancé (who can possibly be a part of the adoption of your child even though you are not yet wed; check the laws in your state) takes on full financial, etc. responsibility, your ex-husband isn't going to be absolved from paying back child support.
Get thee to a lawyer. Get your divorce and child support agreement altered to garnish your ex's wages and possibly even repo some of his possessions in order to pay the old debts. They will not be discharged by an adoption. This is not like declaring bankruptcy.