Flight Crew Air Transport Pilot (ATP); Flight Crew Multiengine; Flight Crew Instrument
Confusion; Communication Breakdown
ATC Issue All Types; Deviation / Discrepancy - Procedural Published Material / Policy
Air Traffic Control Provided Assistance; Flight Crew Requested ATC Assistance / Clarification
Airspace Structure; Human Factors; Procedure; Staffing; Environment - Non Weather Related
Upon takeoff on XXR at ZZZ; we were flying the ZZZZZ's 3 departure procedure. Usually Tower promptly switches us to Departure but there was a delay and the PM queried the Tower Controller for a switch. Tower did not immediately respond; roughly 10 seconds later Tower issues us a change to a departure frequency that was not what was in our clearance. The PM switched to that frequency but it turned out to be a Local Tower; we then immediately switched to what was issued in our clearance which turned out to be the correct one. Approach asked us why it took so long for us to switch and that they were looking for us on guard as well. Nothing else was asked and the flight continued uneventfully.The ZZZ Controller was working both Tower and Ground at the time of our departure and it was moderately busy. I think Controller became confused and issued us the wrong frequency. I assumed it was a last minute change by Approach.Make sure local towers have 2 controllers on duty to split up Tower and Ground frequencies. The return of air traffic is happening quicker than the FAA realizes and they need to staff accordingly. We should have asked Controller to confirm the new frequency but the Tower frequency was quickly becoming congested and hard to get a word in without delaying our switch to Approach even longer.
After takeoff; there was a delay in Tower sending us to departure frequency. It was clear Tower was working more than one frequency. When I could; I queried Tower if they wanted us to contact Departure; but received no reply. Approaching 4;000 feet (our initial level off); Tower directed us to contact Departure on XYZ.A; rather than the XYB.CD on our PDC. While I was inputting the frequency; I heard someone calling us on guard and I missed the last digit of the frequency. I checked in on XYZ.E; then immediately switched to XYZ.A and checked in. It was approximately 5-10 seconds after Tower giving us the switch until the check in. Two issues: 1) ATCT was working more than one frequency causing delays and distractions; 2) The departure frequency in our PDC was different than that depicted on our assigned departure.While in cruise I reviewed the ZZZZZ3 departure (which we were assigned) and noticed that frequency is reflected as the departure frequency. If I had noted the discrepancy prior to departure; I could have verified which ATC was expecting (depicted or cleared via PDC).
Flight crew reported that during takeoff they dialed in an incorrect frequency while trying to contact Departure.  The crew attributed the error in part; to the Tower Controller working both Local and Ground positions.