Husband’s papers are still processing; is he going to be allowed to travel within the US?

No one here has enough information to answer your questions. You need to have an in person consultation with an experienced immigration attorney, if you want a second opinion. Do not let geographic restrictions get in the way of retaining the best attorney. Pick the best attorney you can find and remember one rule: a good attorney is generally never cheap, and a cheap attorney is generally never good so don't choose based on price.

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Is he already out of status? And if yes, was he out of status more than 180 days before filing for adjustment of status? Although filing for adjustment of status stops the accrual of any unlawful presence, it does not give anyone "legal status."

We would need more facts on his status to advise.