I'm in the process of a divorce. We have been separated for 3.5yrs living in two different states during this time. I've been the primary provider for our 3 children since birth. They reside with me and only visits him in the summer when I take them and pick them up. I've filed for a divorce and full custody asking for child support only because the courts require it. He does not support my children now nor do i suspect him to in the future. My soon to be ex does not work and claims not to have any bills. He lives with his family and when he was served papers He decides he wants shared custody. I know it is only to keep him from paying child support. He hardly speaks to them and when they do visit him over the summer they spend most of their time with their cousins. I've repeatedly asked him to call them more and work on his relationship with them not me. That didn't go to well. I want to fight this shared custody only because i know he's not doing it for the right reasons. But at the same time all i wanted was for him to be apart of their lives so that they know they are loved by both of us and not just me. Not sure on what my next steps should be. I want this divorce finalized, but i want to do the best thing for my children.

States have different options for custody sharing. It sounds as if you have a good case for primary custody, which in some states requires that the other parent provide child support.
Whether or not he comes through with that financial assistance is a separate issue. But he shouldn't be barred from seeing his children.