Air Compressor Pressure Drops During Operation

Good day,
I am having an issue with an air compressor in our workshop. The compressor reaches the required tank pressure normally when there is no demand, but when we operate pneumatic tools the pressure drops and the compressor cannot maintain it.
I have checked for obvious air leaks and the intake filter is clean. The compressor motor also appears to run normally. I am not sure whether the problem could be with the unloader valve, pressure switch, check valve, or simply insufficient compressor capacity for the tools being used.
What would be the proper way to troubleshoot this? Should I first measure the actual CFM output and pressure while the compressor is under load, or check the pump and valves first?
Any advice from someone experienced with compressor troubleshooting would be appreciated.