My ex-husband is paying child support but has no contact with our 2 young children; can I file for abandonement?

You should have a lawyer help you file an application for permission to move out of the state.  I think the biggest problem you're likely to have with this is getting the papers served on your ex, and that will be something where the attorney's expertise will justify the cost.

These kinds of applications used to be much more difficult, but courts are now quite properly giving more recognition to a divorced parent's constitutional right to live where she pleases -- and the reality that the custodial parent's quality of life makes a big difference in the children's quality of life.

In the typical case (where the other parent objects), there has to be a new visitation schedule set because alternate weekends won't work, the responsibility for the extra travel has to be divided up, etc.  Here, assuming he doesn't object, I'd expect the court to simply order "liberal and reasonable" visitation, and leave it at that.