PA-28 flight instructor reported a runway incursion during taxi while another aircraft was on takeoff roll at a non-towered airport.

Narrative

This incident occurred at ZZZ. Aircraft X Crossed Runway XX while another aircraft had started their takeoff roll. After a full stop landing on Runway XX we exited the runway and refueled at the self-service station. While taxiing back to the FBO we crossed XX-XY on Taxiway 1 and were holding short of XX-XY on Taxiway 2. There was an aircraft on base that had just made a call that they were turning final; I had them in sight and decided to expedite my crossing but could not see well down to the approach end of the runway due to sunstrike. At that moment an aircraft had pulled out onto Runway XX and began their takeoff roll as I was crossing and still on the runway. I did not hear the aircraft make a radio call that they were departing. After getting clear of the runway the pilot of the aircraft taking off asked me over the radio if I saw him on takeoff roll; I replied that I did not see him and immediately apologized. After returning to the classroom I reviewed the audio recordings of what had just transpired. The aircraft did make a radio call that they were departing Runway XX at XA:37:20. I made my radio call that I was crossing XX thirty seconds later at XA:37:50.