Is giving up your parental rights the answer to your problems -- maybe, maybe not.
Does it matter to you if mom gets a new boyfriend and tells your kids( )to call him dad? What if new dad adopts your kid(s)? What if kid(s) change last name to something other than yours? These are all things that can happen if you give up your parental rights.
If your parental rights are terminated, is your child support obligation is also terminated? That is often true. But if mom is getting public assistance, they will probably not let you off the hook so easily.
If what you really want is to pay less child support and stay involved in your kid(s) life, mom can agree to lower your child support payment. Ask her if she will do this.