CRJ-700 Captain reported a 'last minute scheduling change' at different gate that was already 'boarded up'. Crew completed a safe expedited departure. Crew received airborne notification of a manifest error. Flight release was corrected on return flight.

Narrative

After a delayed flight from ZZZ1 to ZZZ I was called by Crew Support and asked to get to a nearby gate as soon as possible to operate the flight from ZZZ to ZZZ2. From the time I shut down my previous aircraft to the time aboard the new aircraft was less than five minutes and I was the last person aboard the aircraft. I briefed the crew quickly as we were already boarded up; and we dealt with the challenging situation of operating without an APU. With so much on my plate I had no chance to verify the crew and had to rely on Crew Support and the Gate Agent to verify that the crew's ID's had been checked. About midway through the flight the Dispatcher discovered that the FA [ID No.] was on the release and Crew Support had assigned a different FA [ID No.] to our flight without informing the Dispatcher. The release was correct on the return leg.The causal factor for this incident was a last minute scheduling change due to major crew shortages leading to confusion and loss of situational awareness in crew assignments. Contributing factors were individual last minute crew assignments at short notice that prevented sufficient redundancy in preventing flight release errors.Sufficient reserve staffing on non-holiday holidays when individuals are more likely to call in sick to prevent scrambling for crews to fill flights.