Let me try and make a few quick points, some of which have already been made but at least one of which is new.
1. On your 2021 tax return, just claim your children. If they lived with you more than half the nights of the year, you don't need any special forms or paperwork. If the other parent doesn't claim them, the IRS won't do anything special. If the other parent also claims them, you may need to show the IRS proof of where the children lived. The IRS would send a letter to investigate and you would send your proof in reply to that letter; don't send proof unless you are asked.
2. By not filing a 2020 tax return, you missed out on $1100 of stimulus payment/recovery rebate per child that you could have gotten for your children, even though you were not working. (Unless you already got a stimulus check for the children.). You could file a return to get those credits, but it would cause the IRS to start investigating why you and the other parent both claimed the children. (The other parent got that credit and would have to pay it back if you can show the children lived with you, and you did not give the other parent a form 8332.)
3. By not giving the other parent a signed form 8332, you were facilitating his possible tax fraud. He might have been able to qualify for EIC or head of household status that he would not be eligible for if the children did not actually live in his home more than half the nights of the year. This will probably not cause you any grief, and the IRS is too busy to investigate unless you filed a duplicate dependent claim for those tax years, but it is something to bear in mind.
4. We can't really help you with your landlord since it is not a tax matter, but there are legitimate cases where a parent might not claim the children who live with them as dependents. If you are trying to get a rent subsidy based on family size, you will have to discuss with the state or city agency, what other proof might be acceptable.
