"Sheltered Housing" in American English?

Solution 1:

When I (an American) hear sheltered housing, I think of homeless shelters.

What you're talking about sounds like a mix of different types of organizations that we split up into different categories: disabled housing, retirement homes, psychiatric hospitals, public housing, etc. I don't think you'll find one word that encompasses them all. It's not that the concept doesn't exist over here, it's just that we don't really have an all-encompassing word.

The closest thing I can find is assisted living, which might be what you're looking for. (Edit: I just saw that you already mentioned this term, but hopefully the previous paragraph helps.)

Solution 2:

What you are asking about is on a continuum between independent living and long term care.

This paragraph from this link may explain why the exact portfolio of services in your government's term is not established in the US:

These facilities may go under various names, such as nursing home, personal care facility, residential continuing care facility, etc. and are operated by different providers.

While the government has been asked with the LTC (Long term care) industry not to bundle health, personal care, and services (e.g., meal, laundry, housekeeping) into large facilities, the US government continues to approve that as the primary use of taxpayers' funds instead (e.g., new assisted living). [Wikipedia]

Solution 3:

A more common North American term might be group home

A group home is a private residence for children or young people who cannot live with their families, or people with chronic disabilities who may be adults or seniors. Typically there are no more than six residents and there is at least one trained caregiver there 24 hours a day.

In Canada we use the idea (and probably meaning) of "sheltered" when we speak of sheltered workshops for the disabled.