Upvote:0
For a domestic flight within the US, the airline staff establish your identity through your documentation. This can be something like a US state drivers license, or an international passport. Airline staff are not trained immigration officers, and won't even look for a valid visa in your passport. As long as your passport is valid, it should be perfectly acceptable.
If you have a US state drivers license, that's an even easier way that does not involve your passport at all.
Upvote:2
Yes, you can fly domestically within the US.
Although you are not out of status, you should read the questions on this site concerning domestic travel by those who are. The answers make it clear that your concern about being asked for proof of immigration status is likely to be ill founded, except in some well defined cases where you are likely to encounter Border Patrol officers.
There are some questions here about traveling domestically in the US while being "out of status." These cover the likelihood that you will be questioned about your immigration status, which is extremely low unless you're flying through McAllen or Brownsville, or perhaps another airport within 100 miles of the Mexican border (it's also possible near the Canadian border, but I haven't seen evidence that it actually happens):
You are better situated than the askers of the first two questions, and possibly of the third, because you are actually in the US legally. To summarize: