You can show any state ID to TSA officers, airline employees and if necessary police offices. In fact, it's better to show them a state ID since almost all passengers show them a state ID. However, if you are not a US citizen or US permanent resident you must always carry your passport with a valid I-94 with you. (If you are a US permanent resident you must always have your "Green Card" with you.)
See 8 U.S.C. § 1304:
(e) Personal possession of registration or receipt card; penalties
Every alien, eighteen years of age and over, shall at all times carry
with him and have in his personal possession any certificate of alien
registration or alien registration receipt card issued to him pursuant
to subsection (d) of this section. Any alien who fails to comply with
the provisions of this subsection shall be guilty of a misdemeanor and
shall upon conviction for each offense be fined not to exceed $100 or
be imprisoned not more than thirty days, or both.
Your alien registration card is your I-94. In practice, visitors are very rarely asked to present their passport and I-94. In most states, only Customs and Border Protection agents (as well as other federal law enforcement agents) can demand your immigration papers; police officers cannot (possibly, Arizona is an exception).