You do not need an address in the US to sign up for Social Security. You do need to provide a physical address in the DR for your initial confirmation notice and a mailing address to receive your annual form to prove your are still alive. I use my friend's P.O. Box in Puerto Plata. But my original confirmation was actually delivered to my physical address in Costambar by the Dominican Postal Service. A Postal Service worker came on motorcycle and actually handed it to me! I almost fainted. Yes, 65 is the age for Medicare. Just go to the website to review your options. I did not sign up for part b because I will not go back to the U.S. for any healthcare. I'll just receive my treatments here. I am happy with my doctors and my healthcare coverage here and the pharmacy is good at obtaining my medications for a blood cancer (incurable but treatable) and Parkinson's.
Are you aware that every year you need to fill out forms when you live out of the US? If you don't comply you could be terminated. Check with SS on this because rules change all the time. But I know this was in force some years ago because I checked myself.