Selling home, what to do with mail? What is his "home address" now?

Questor, when my Dad moved in senior living, since I was his Power of Attorney for health issues and financial issues, all of Dad's mail was forwarded to my attention.

Dad already had me as a signer for any checks that needed to be written from his savings/checking account. New checks were made saying as an example John Smith OR Jane Doe.

You can check with Accountants to see if they do this type of financial management, but there will be a hourly or set fee.

By the way, wearing diapers isn't a health issue. There are people who are in the working world who wear Depend type products.

For Social Security and Medicare purposes he may need to use the address he is living at. Same with his ID and voting registration. This is his legal address.

I had all Moms bills and bank statements routed to me since I was the one paying her bills. I only had one problem, the hospital admitting. The argued they needed her ALs address to send her bills to. I argued they needed to come me because I paid the bills. I wasn't winning that one. So then I said well then you must put Moms mailbox number on the bill. If not they hand the mail to Mom and since she has Dementia that bill could go anywhere and then they don't get paid. I later had to call their billing office about something else. Found out in talking to the clerk that I was in my rights to have the bills come to me, that the admitting person was wrong.

You can hire a Geriatric Care Manager in California to manage his care.

https://www.agingcare.com/local/geriatric-care-managers/ca

My husband is POA for his brother, and we fortunately live in the same state. Any official documents that my husband needs to handle come to our address, with brother's name c/o my husband. It's not at all uncommon for people to have a residential address and a different mailing address (though that's more usually the case for peple who have PO boxes). So all the bank statements, pension statements, and even social security statements come here. He has supplemental health insurance, and that is based on the residential address zip code, but that mail also comes here. For other purposes, such as state id and voting, he'd have to use the place where he resides, but even for those, it might be possible to have a different mailing address c/o whoever is handling his finances, etc.

If you are already POA for health care and POA for finances (they are separate documents that hopefully are in place already), then a licensed fiduciary in CA can assist you. If you don't have these, I think you need to locate an elder care attorney and plan a trip to CA so you can get your brother to attorney office to create and sign those documents.
A geriatric care manager can help with a lot of these tasks, and a fiduciary may be needed for some steps. Usually a geriatric care manager has a relationship with a good fiduciary, can recommend good attorneys, etc. I do know a couple of resources in San Mateo county, if he is there or near there.
There are MDs who specialize in seeing homebound elders, which may be a better fit for your dad at this stage.
Good luck.

After dealing with statements and info I needed from my mom when she was in an AL , that she often lost or couldn’t remember… when I moved her here, to memory care , I changed her mailing address to mine with C/O (my name) I am her POA.

There are fiduciaries you can hire. They charge approximately 90-100 an hour. They pay bills, manage everything from phone to placement if you ask. You kind of tell them what you want handled. You need to be certain that they are Licensed trained fiduciaries, not financial managers who "call themselves" this. Call and elder law attorney in the area where your Dad is as these Fiduciaries are used by the courts to help with elders who are alone and need help; they often serve as guardians for the elder through the courts.
Now on to the address. I was POA for my brother and Trustee of Trust and did all taxes, wills, accounts, bills, etc. Everything financial. Because of this he had two addresses really, but the mail that came to me I arranged with each and every entity to have sent to me (wow what a process!). I lived in No Cal and he in So Cal. It made it much more difficult than had I been able to drop in and get his mail.
So the dilemma is, yes, where will Dad's mail go. It may be best, if it goes to the POA or whomever is his guardian. But it can't be changed for Social Security to do that and in fact is very difficult to change anything at all with social security. If Dad is competent he can tell the B and C that they need to send certain pieces of mail to his POA. If not it is harder.
A tough dilemma. I honestly don't know what to tell you for your Dad. It was hard enough always to do for my brother. I wouldn't try it again.