If the father of our child has full and sole custody and the mother no longer wants to pay child support because she does not see the child is she still obligated to pay support after?

First the mother can not just relinquish their rights, if it was that easy, there would not be 1000's of dead beat parents in jail now for non payment, they would just give up their rights.

So unless the other parent is remarried and has a spouse that wants to adopt, the non custody parent just has to keep paying support. Not seeing the child is not a issue.

Also full and sole custody does not mean there can not be visits, if the non custody parent wants visits, ( unless they were a danger to the child) they have to get visitation in court