Is LTC covered by NYS Medicaid? My mother had been in AL for 3 years, had several falls and was hospitalized and released to rehab recently.

Medicaid actually does allow you to have assets and receive LTC. You can have a house for example. You are in NY which has some of the best Medicaid programs in the country. You can $15k in cash assets if you are single or $22k for a couple. You can have a house worth $800k and a car. And if your moms monthly income exceeds the limit for LTC Medicaid then there is a program where she can pay her excess income to the Department of Social Services & she’ll qualify for LTC Medicaid.

Medicaid usually pays for LTC but only when the person has no assets. What assets Mom has will need to be spent down to the Medicaid cap (2k in my state) Some of her money can be used to preplan a funeral.
I would make an appointment with Medicaid and sit down with a caseworker to find out the criteria Mom needs to meet. Do it now, do not wait until her money is almost gone. Where I live, you have 90 days to spend down, find a LTC facility, and provide all the paperwork needed. Like 5yrs of bank statement. I lucked out that Mom had kept all her statements for that long and statements showed pictures of the checks.