My wife and I had a similar situation. Her GC and drivers license are with her new name, but her passport is her old name. We haven't had any issues at all traveling, both together and her solo. She has re-entered the US 3 times with this situation. Twice through the airport in Houston, TX and once on a land border crossing in Brownsville, TX. She carries her GC, passport and a copy of our marriage certificate with her...although she has never been asked for the additional documentation.
I wanted to update this thread with our recent experiences. My wife & I decided to not get her DL until she gets her EAD, that way she can get 1 year of validity. So, as I still intended for her to have a form of American ID, we decided we would temporarily focus on getting her a TX ID Card.
Hence, on Feb 6, we went to a local TX DPS to do that. They wanted 2 pieces of mail from different sources addressed to her name, between 30 & 90 days old. Since her POE was just Dec 31, she didn't have that. So they allowed me (someone who lives with her) to sign a waiver. So they basically wanted documents from me instead. I didn't have 2 documents on me; only 1. Needless to say, after we burnt 1.5 hours waiting, we got nothing.
So yesterday was Betty's Biometrics. We decided to attempt the TX ID again as well. This time I brought 4 documents, as I'd rather have too many but not enough. I brought a paystub received between 30 & 90 days, 2014 W-2, my current auto-registration & the one that goes into effect on March 1, my current insurance policy. They accepted those. That said, they advised us that her TX ID would only be valid through the date the Visa is valid (essentially the same as the I-94, as the Visa's validity end-date changes once the beneficiary POEs to 90 days after POE), March 30.
So basically we consider it all wasted time, essentially. Once she gets her EAD, we will go back to TX DPS to renew her TX ID for one year & get her TX DL, which will also only be valid for one year. Then when she finally gets her GC, we will renew both to the normal ones (6 years for DL IIRC & 8 years for ID IIRC). I hope this post helps others in TX.