when I did a continuity test and take one probe for continuity on the Pin 1 XLR jack and the other probe to the black cable to the power supply lug, it beeps. It also beeps at various places on the pcb. Is that a pin 1 problem?
Not necessarily, because an ohmmeter is testing at DC, and the connection should eventually meet up again.
The cable shield pins should be connected to chassis as directly as possible (assuming conductive metal chassis), but the circuit common/reference/"ground" conductor should also be referenced to the chassis. You will still detect a low resistance connection, but if you follow the current path which any interference on a cable shield would take to get to other cable shields or the power supply, the path should not be forced to go across the circuit board.
The XLR is also soldered directly to the pcb…
Depending on how inexperienced the designer was, the easy way to connect the XLR pin 1 is directly to the circuit common (what is usually called circuit "ground") which is the definition of the "pin 1 problem." Confused thinking about the purpose of various different things all sometimes called "ground" is what leads to the problem, any current flow through a conductor will result in a voltage, described by Ohm's Law. If that current flows through the circuit reference, the voltage created now is the reference voltage for the circuitry, which means it will show up directly on the outputs. You want the current from cable shield to be able to flow on the chassis to return to the source via other cable shields, or power supply lines (through filter or parasitic capacitance), or the safety earth conductor. If you are thinking wouldn't the voltage still show up on the outputs because the circuit reference is still connected to the chassis? The answer is yes, but the entire reference will be at that voltage, there will be no difference at different locations of the circuit, so it becomes "common mode" noise, i.e. the same on both legs of a balanced connection. So balanced connections are required for lowest noise, since an unbalanced connection has no way to reject the common mode noise.
There is a way to create a PCB layout without pin 1 problems with PCB mounted connectors, but it requires having a copper pour around the connectors with only pin 1 connections, and a strip of metal connecting that copper pour to the chassis (a wide but short length strip rather than wire so the inductance of the connection is very low), and the rest of the circuit ground only connecting to that shield pin connection at one spot. If the XLR connectors are the kind with four pins, a separate pin for the shell, then just connecting the fourth pin of all the XLR connectors to that copper pour may be enough.
If the chassis is plastic, or anodized aluminum, or some other non-conductive construction that may be all you can do.
But that is more a design issue than a modification suggestion, I don't think it would be very practical to try to make modifications on an existing design with PCB mounted connectors. Probably in that case using the Neutrik EMC enhanced connectors would be the best thing to try.