Ok, there are four issues here.
1) enforcing order already in place
2)possible change from joint to full custody (NJ having custody)
3) Child Support Amount
4) Child Support Arrears
#1 and #3 is what you are going for. File a contempt ASAP. This has nothing to do with child support and nothing to do with changing custody. Simply file contempt on the fact he's not getting child every Tuesday thru Thursday.
File to have a change in child support due to change in circumstance. Change in circumstance is that your husband lost his job.
Right now custody agreement is joint, it depends on both incomes. When there is joint, if NJ made more $ she may end up owing your husband, and vice versa. Since they are both unemployed-it may even out.
This is a separate issue than his back child support-HOWEVER, the sooner you have it modified the less he will owe in back child support.
#2 and #4 are NJ's issues. If she wants joint changed to full custody on her part, she needs to file. I would hope that her apparent lack of best interest in child, judge would not look kindly on making him EOW dad since she won't allow him access to child as it is.
As for Child Support Arrears-she must go thru the child support enforcement agency to show he is not paying. If garnishment doesn't automatically come out of his check(as it seems it isn't) they may not be keeping track of what is owed/paid. Since you haven't heard anything thus far, unless she causes a fuss about it-odds are you may not. But I hope you can prove every cent you gave. And from this moment forwards, do not give anything that is not thru the child support agency, or make NJ sign a receipt for every penny.
Your income(as new wife) will not go into child support calculations. The main exception to that is if he is stay at home dad with your kids and you are breadwinner, then the judge may impute his income (meaning give him atleast a wage of minimum wage).
And let me make one thing clear. He is not trying to get his visitation-he is trying to get his parenting time. He is not a visitor in his child's life, he is a parent. They have joint-he should be making efforts to be involved, despite NJ's husbands' antics. He should be calling child's doctor, going to visits, calling school, volunteering at school. There are many ways to get around not seeing child just because NJ won't let him.