Marriage
You could get into some binds down the road without that paper. One example: if either of you dies, the survivor needs proof of a valid marriage to collect Social Security and pension benefits.
Separation and Divorce
It's hard to think right now, because separation is traumatic. Do you have enough money to get by? If children are involved, and he has not provided any money for their support, you can file a complaint for child support. If you've been working only part-time, you may need to file for a limited divorce, asking for alimony pendente lite (pending a final divorce) until you get a fulltime job.
Time with the Kids
No, you can't do that. It's hard to meet the children's needs on your paycheck alone while the overdue support piles up. But the law doesn't link support payments to visitation. Your ex-wife is still the children's mother, and "punishing" her by keeping them apart hurts the kids. But you can ask the court to enforce its support order by garnishing her wages and other means.