I need help and have no idea where to turn to. Are there government programs that could help?

Please contact a State social worker. Could you elaborate a little more? You say he can't afford the facility he is at, how long has he been there and did he take his finances into account when he moved in? What is it that has changed? Why can't he afford it now but could before? It sounds like he may be stuck in between not being medically qualified for LTC Medicaid but at the same time can't afford assisted living. A state social worker is going to need to help find him apartments under HUD where they take a set portion of his income for rent and also find him support for his medical needs.

Dear "MyAgingFather,"

I'm so sorry you are finding you and your dad in this situation. Does your state have a local "Area of Aging Agency" where you can speak with a social worker? Or can you get a Placement Agent who knows all the types of facilities and who has the connections/rapport with various places/people where you can tell them your specific situation and they can tell you what is available.

You need a jumpstart as I always say one resource leads to another until you get the specific type of help you need. Whoever you talk to, if they don't have a solution always ask them for another resource and usually they are able to provide them.

I pray you get the help you need and very soon!

Have you spoken with the Administrator of the place he is at now?
If you take him home, how often does he end up in the hospital you mention.
Generally, what you have to do often enough is refuse to accept him home from the hospital. They cannot force you to accept him back into your home; Tell them that you are neither mentally nor physically capable of taking care of him in your home now and he requires placement. Tell the Social Worker this the first day of admission to either ER or Hospital. Don't listen to their arguments about how he doesn't need it or they can help you with more help and make it work. That is just an effort to get you to take him home.
If they are claiming "he doesn't need it" I am uncertain what they mean. They MAY mean that your father is mentally competent enough to make his own decision where he wants to go. If that is the case tell them, and tell HIM that coming to your home is not an option, and if they discharge him anywhere unsafe it is an "unsafe discharge". The burden is on them at that point. It is basically what most Hospital Social Workers spend their time doing.
There are some government programs for cheaper rent, but you seem to indicate Dad won't thrive on his own, and there is really no program that will provide him in home help. There are huge waiting lists for subsidized housing.
This is a huge problem in our country.. You are not alone. I am so sorry.

Sounds like your Dad should not be in independent living. People with a Dementia need 24/7 care. Dad needs a LTC facility that takes Medicaid. Hopefully you have POA it will make things much easier.