From your previous posts you're a Kiwi so you will have to cancel the USCG paperwork. Not to be nitpicky but to avoid confusion, in the US this is correctly called USCG Documentation
is a different procedure that is done in individual states. Depending on the boat and circumstances a state registration can be instead of or in addition to USCG Documentation
Either way, non-US citizens are not allowed to Document a boat (you can state register but that doesn't seem to be relevant in your situation). There is a process with the USCG to cancel the documentation
so you can register the boat in another country.
As far as exiting the DR without problems, it has been years since I have sailed that area but I have never had a problem entering or exiting anywhere in the Caribbean
on a boat registered or documented to someone else. If you have a problem changing ownership, title and registration in the DR just get a letter (preferably signed and stamped with lots of official looking stamps and seals) from the owner giving you permission to sail the boat and check out.