ok. so if you have a new registrar, then there is a registry-registrar conversation here bound in an ICANN contract. You really need somebody in the new registrar, talking to the old registrar AND the registry, who is motivated to push buttons (metaphorically) to make the old registrar want to give the data up. I am pretty sure that the names, and associated records, are "yours" if its "your" zone. The contract for services outside of DNS which may lie behind those names "let us host your web for you" are different: you could wind up with your DNS zone, and your web offline because they manage that too. Not all "hosting" is DNS only.
If you have a new registrar, this situation shouldn't be unfamiliar to them..
If you have a new registrar, this situation shouldn't be unfamiliar to them..