James, if you are flying a Cayman flag, you are required to have obtained a US Cruising Permit
BEFORE entering US waters and the penalties for being here without one can be quite steep. (I assume you are not under diplomatic status.)
If your vessel does not have foreign flag status, you can register it as a motor vehicle with the NYS DMV (Department of Motor Vehicles) and that will allow it to be used as a domestic motor vehicle in all 50 states--IF and only if you are a NYS resident during that time. You'd have to maintain a residence in NYS to qualify.
Your choices are to register the boat as a motor vehicle--in which case it is subject to different laws in each state, as well as taxation for sales/use tax when registered--or to foreign-flag it, in which case you must have the cruising permit
and that is good for one year, at which point the vessel must leave the US (I think for 12? days?) and then re-enter with a new permit.
While you can probably get away with registering the boat in NYS while you live here, and then simply maintaining the NYS registration while you wander around, you will have to pay sales/use tax on registering the boat, and technically, if you are in any other state for more than 30-90 days, you have to re-register the boat as a resident of that state. The term varies--but each state requires vehicles to be registered locally if they are there "long term". If you carry insurance
, that will be affected by this. And some marinas
and YCs will not allow you to tie up without liability insurance
So...It all depends on which way you want to go. Foreign flagged, state registered, either way you will have issues as an 'alien'. And bear in mind, if your paperwork is improper, the vessel is subject to arrest (seizure) as well as monetary fines.