Yes, you can fly domestically within the US.
Although you are not out of status, you should read the questions on this site concerning domestic travel by those who are. The answers make it clear that your concern about being asked for proof of immigration status is likely to be ill founded, except in some well defined cases where you are likely to encounter Border Patrol officers.
There are some questions here about traveling domestically in the US while being “out of status.” These cover the likelihood that you will be questioned about your immigration status, which is extremely low unless you’re flying through McAllen or Brownsville, or perhaps another airport within 100 miles of the Mexican border (it’s also possible near the Canadian border, but I haven’t seen evidence that it actually happens):
- I’m out of status but want to travel from Los Angeles to New York, USA
- Traveling inside the U.S. with a tourist visa from 3 years ago
- Expired visa, can I still fly inside US without risk?
You are better situated than the askers of the first two questions, and possibly of the third, because you are actually in the US legally. To summarize:
- Airline staff are not interested in your immigration status. They just want an ID document that matches the name on the electronic ticket.
- You mention Customs and Border Protection. You will not encounter CBP officers during a domestic flight.
- TSA (security) staff are not interested in your immigration status. They are not trained to determine immigration status and they will not look at any page in your passport other than the one with your picture and name. I can attest to this from personal experience flying domestically in the US using a foreign passport for identification.
- There is a chance when you travel in some airports that Border Patrol officers will be working at the TSA checkpoint. Border Patrol is one of the “immigration police” forces (the other being Immigration and Customs Enforcement, or ICE). Only these officers will be interested in your immigration status. If you have a pending adjustment of status application, you should be fine.