You don’t need to have a legal status to travel. You need a state issued ID or driving license as others have mentioned before. Regarding renewal of passport it does not matter if it’s expired or has few months remaining, you can renew it.
I personally went to NYC consulate and did the renewal process(in 2018) it took under 1 hour.
You will have to leave a prepaid envelope. Once the passport is ready the consulate will mail you your new passport using that prepaid envelope.
You can also have the passport expedited its going to cost a little more money & few extra steps but if its not urgent you should get you new passport in about 1 month.
Again that was my personal experience from 2018 things might be different now post covid.