After approval of your deferred action application, you should be able to receive a driver's license from the state you reside in. Most domestic airlines do not require anything supplemental to a valid state ID to board the plane. So, the answer is that, most likely, you will be able to travel within the country.
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You can travel within the US only after Deferred Action has been approved.
J Charles Ferrari Eng & Nishimura 213.622.2255 The statement above is general in nature and does not constitute legal advice, as not all the facts are known. You should retain an attorney to review all the facts specific to your case in order to receive advise specific to your case. The statement above does not create an attorney/client relationship. Answers on Avvo can only be general ones, as specific answers would require knowledge of all the facts. As such, they may or may not apply to the question.
As you stated, you are allowed to travel within the U.S. even in unlawful status, as airlines employees are only checking some sort of ID (e.g., your native country's passport) to make sure the name on the airline ticket matches your ID, and the picture matches you.
However, once you fly, you do expose yourself to a higher risk of encountering some airport security personnel or even DRO or CBP officers who may then investigate and detain you. But it is not correct to assume that airport security will automatically stop you just because you present just an airline ticket and a valid foreign passport for travel within the US.