"do I have a claim to either boy as a 2020 dependent?"
No.
For children of divorced or separated parents, where the children lived with both parents more than half the nights of the year and both parents meet the basic qualifications to claim the children as dependents, the first tie breaker is which parent did the children live the greater number of nights. It sounds like the children lived with you about 240 nights (8 months) but lived with the other parent about 330 nights (11 months).
The other parent can release one or both dependents to you, but the IRS does not require it. If a court orders it, the IRS will not follow the court order, but will wait for a release form from the parent who has more custody time. If you don't get the form, you can sue the other parent in family court but the IRS doesn't follow state court orders, only its own rules.
For 2021, if you have custody more days/nights than the other parent, then you are the parent who has automatic first right to claim the children on your 2021 tax return.
All the rules are here.
https://www.irs.gov/pub/irs-pdf/p501.pdf
https://www.irs.gov/pub/irs-pdf/p504.pdf
