I've detailed this several times before here in the forums and in my book
The Anarchist's Guide to Travel (shameless plug), but here's the easiest way to show proof of residency in the USA.
first, go to a print shop like staples or office depot and ask if they have a generic lease agreement form you can copy. many will, although nowadays it's probably easier to find one online. either way, sign as the leaser, and have someone else sign as the landlord (so the signatures are different) in any names you want with any address you want. boom. proof of residency. it's stupid easy and i've been doing this for most of my life when the circumstances called for it.
now some states will only give you a temp id and mail you the real one, so you might want to make the lease out for a friend's address or somewhere you can receive mail. other than that, you should be fine. if you need a birth certificate, you can easily get one mailed to you through
Vital Check. I've used this service multiple times when i've lost my birth cert through various travels, and on average it cost me about $25 to get a new one mailed to me. of course you'll need to know your SSN and your parent's info (sometimes their SSN's as well), but that really shouldn't be too hard.