Nice to meet another oldie interested in HUD senior housing! I love living where I can walk to the neighborhood stores. There are several people in my building who are from NYC but find the South much warmer.
There are a number of websites that list HUD elderly housing by state and city. HUD housing is always affordable because the senior communities only charge 30% of income for rent.
Here is a list of HUD apartments in New York State. Unfortunately it's alphabetical by name of unit rather than by city, but it does indicate whether a property is "elderly" or "family" or whatever.
https://www.hud.gov/sites/documents/DOC_13030.PDF
SPM manages a senior community in Flushing:
https://spm.net/communities/details/...layout=listone
Unless you have your heart set on NYC, there are senior HUD apartments in Florida and other southern states. Some may have very long waiting lists, though.