DIY

It might help if you explained why you think that is the preferred term. I'm admittedly ignorant on the subject, but I don't see a problem with OP's wording.

"Unhoused" is the preferred nomenclature.

"Unhomed" is the wrong tense and way to use when referring to someone in this situation.

"Homeless" makes it seem like they don't have a home, which isn't true, as homes are not always a structure and can be considered more of a concept.

Unhomed is past tense and still refers to them not having a "home", which again, is not true and is generally seen as a demoralizing way to refer to their situation as if they don't, or can't build a home from the communities they find themselves in.

Unhoused is referring purely to the structure of a house, not the concept of a home. A house is a physical thing, whereas a home is where you make it, sort of thing.