JSalas, Davdah is correct about flight diversion.
If you have your passport still I would suggest in taking that too. Your immigration docs you must take, to prove status.
My plane coming back nearly diverted elsewhere as our plane had mechanical problems and was losing pressure or something and we had to fly at a unusal height. Oxygen was not that good on plane I can tell you, rather frightening.
We were half way towards the Mainland of US but they decided it was best after all to turn around and go back to HI.
The pilot apparently was trying to find clearance on where to land that was nearer, so we could have ended up on foreign territory.
I flew ATA..which now no longer run thankfully..at the airport there were 2 other planes that also had mechanical problems, one where the door wouldn't shut properly! eek
However saying all that, domestic flights do not go through the same areas as international ones.
There isn't the usual ID checks like we do entering the US.
I had expired ID, and even flew on a different last name, and it didn't seem to be an issue.
My son flew on his passport as he had no other ID.
We were in the middle of AOS. Our visa's had expired. No GC, expired work permit, and new and old passport at British Embassy.
My husband of course used just his DL.
Airline told me I could just use a DL if I had one, but I didn't.
However as you are a Citizen of another country and not of this one, I would take a passport if you can and your immigrations docs.
Alternatively..I believe you can apply for AP under TPS?
You don't need it, but I guess maybe handy if you did ever divert to another territory.
Something to look into anyway.
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God Bless America - God Bless Immigrants - God Bless Poor Misguided Souls Too

Mr S.U.