Custody is where the children sleep overnight (with some exceptions considered if one of the parents works nights). So having the children "saturdays and sundays until 830pm" will not count as custody for tax purposes, and "tuesday after school, fridays after school" only counts if they stay overnight.
It doesn't really matter what the new spouse says, just what the IRS says. If your ex files a tax return claiming the children as dependents, the IRS will eventually send letters to both parents to start an investigation. You will need to decide what your position accurately is (it sounds like you can't claim to be the custodial parent) and then prove your position by mailing proof to the IRS. Proof might be:
•The form 8332
•copies of letters from school showing that the bus drops them off tuesday and picks them up wednesday morning to prove overnight custody
•letters from the school, doctors, or other service providers showing that the children get mail at your address.
I would politely ignore anything the ex or new spouse says, and wait to see if the IRS starts an investigation.
