Upvote:2
You can safely use your passport - as its used to check your identity only; and of course there are no immigration controls within the US.
You can also safely exit the US (there is no exit immigration in the US) - but of course you cannot come back in if your visa is no longer valid.
I would try to get your status normalized though, simply because this will help you in the future if you ever decide to visit the US again.