“Should the father still pay child support if the mother met a man and moved to another state leaving the children with the father?”
Orders are good until modified. Though an argument can be made in a support enforcement case that support should have been suspended during a period of support obligation because the child resided with the supporting spouse it would be best to change the order now that circumstances warrant it. You should seek a modification of the timesharing and suport orders as soon as possible.
An order in the hand is worth at least two clever arguments in the future.