To transfer ownership in the USCG records, you need to provide the USGC an iron-clad (notarized) bill of sale, and, if it applies in your State, proof that the vessel's registration has been transferred to the new owner. It may help down the road (when you kark-it and your spouse wants to sail off to the horizon) if you declare both of your names as the boat's owners - you'll have to tell the USCG who is the primary owner and who is the secondary owner (I can't recall their exact terminology).
My experience: I bought my 32-3 (documented vessel) from the estate of member of our club. He'd bought the boat a few years earlier from the original owner who got it documented when he retired from the Great Lakes to FL, but the guy I bought it from never bothered to get it USCG-registered in his name.
To make my documentation name-change task a little more interesting, the original guy had also died by the time I started my process. The club member from whose estate I bought the boat was unmarried with no children and the executor of the estate was his sister (also unmarried without children) - she lived in another city and wanted the boat gone from the estate. Our Bill of Sale was, unfortunately, a bit informal (no lawyer or other 'official' person involved) because our State (TN) does not Title personal non-commercial water craft (or trailers for them!). I ended up paying a vessel documentation consulting company to get me through this minefield, which was a good decision because the USCG took some convincing why there were no State registration (Title) papers for the boat. Even though I could prove I'd paid the State tax on the purchase price and had a valid State waterway use tag, it was a bit of a hassle to get it all done. But, well-worth it in the end, and NOT something you want to put off to some other day, when your records might be harder to find.
However, I still haven't plucked up the courage to request a vessel name change for the boat - still listed with the name chosen by the original owner. There's a fair bit of paperwork, and cost, in just making a name change. If you want to do that, you might consider doing that at the same time as you change the ownership records.
Good luck!