I won't advise you on specifics there, in fact below strategy applies for anyone in need fighting any of the modern "reformed" agencies trying to get us "cases" of their books.
But if your area has Legal Aid for low-income people, or other social-work advocate types **outside** the bureaucracy you're dealing with, get professional advice
Don't be embarrassed to call yourself "on the verge of homeless", you want to qualify for the free consulting.
Some town / municipal / county governments have Social Services people, run meals on wheels, senior center food banks shelters etc.
But ideally from an attorney experienced in such matters.
_______
When you have your stance, position, "legend" in order, about to go settle things with the service-provider agency do **not** feel you need to fully inform them of every last detail.
Not saying outright lie though, not a good idea from a practical POV, besides being able to sleep at night.
Just present yourself and your situation in a way that makes it most likely your case will be approved.
And usually that means **be mainstream, conventional** in your appearance and attitude as much as possible.
Temporarily down on your luck, doing your best to get by,
but **striving to get back** to being a productive normal S&B dwelling taxpaying citizen.
Do **not** act proud of your subversive freedom, do not defend your "right" to be a nomad, etc.
The GPS idea does not fit on the form, nor in the computer database back at the office, big red flag there. You're not fighting for principles here, you are chopping off your feet in order to fit into the normal sized bed (fairy tale?)
WRT to your residential address if you need to go that way
Ideally your ice box is at a stable location with a mailbox, where a "friend" is letting you stay. Get a library card, register to vote get another card, get commercial or government mail sent there to show them, cellphone provider, medical test, whatever. See if you can get your "landlord" to fill in the blanks on a stationary store lease or print one out at the library, open up a utility account there or get it switched to your name. Driver's licence, bank accounts, etc etc.
Obviously bureaucrats just want things to go easy, find out in advance how to spoon-feed them the **info that fits** what they're looking for,
Let them put the data in the right boxes, check check, approve, stamp stamp the paperwork and on to the next case.
If it comes down to it, register as truly homeless, get them to help you get a residential address that way.
Once you've been approved, lasts at least six months or a year, go back to your wandering life as you like. . .