I collect New York child support. My ex-husband is paying "arears" to me. My children are now adults and on there own.

No.

Child support is for the custodial parent to pay for rearing the children. If there were arrears it meant you paid out to compensate. So you are just getting reimbursed.

No the children do not get a share of the money, You can if you WANT to, but there is no obligation to do so. It is to pay you back for all the money you paid out for their care, rent, electric, food and so on