As suggested, go to the Escapees RV Club web site and start reading on domicile. We sold everything and domiciled in Texas for 16 years of full-timing. We didn't establish anything in Texas. We used medical as we traveled. Eventually we had some more serious issues to deal with so our main docs were where we wintered so we'd have time to complete appts. We registered in Texas and set up our Escapees mail service. We also voted national elections using our Escapees address. If you go to Livingston, TX to set up everything that little town knows all about every single process you need to do and you can get it all done in a day usually. If you're asked to sign an affidavit stating that's where you intend to live eventually, sign it. It doesn't hold you to living there. Anyone can change their mind.
The popular states for full-timer's domicile are Texas, Florida and South Dakota. There are a couple more used depending on your circumstances.
If you're on Medicare and have a good supplement for traveling, health care won't be an issue as you travel. If not, then Florida residency is best at this time through ACA.
It's important to break ties with New York before setting up domicile in another state.
We hung out with many full-timers over 16 years and never heard of anyone getting 'caught'.