What is the make/model of the device that has your reverse polarity indicator?
The way many of those reverse polarity indicators work is to wire a resistor in series with an LED from the neutral side of the load to earth ground wire going back to shore. In other words, coming off of the load neutral, there is a path through an LED and resistor to the ground wire and to ground.
Under normal circumstances, since the neutral and ground are tied together on shore, current doesn't (significantly) flow through this path -- there is no (significant) voltage across it, no (significant) potential to drive current flow.
In a reverse polarity situation, what should be neutral turns out to be hot and now there exists 120V of potential between what should be neutral and ground, which drives current flow through the LED and resistor.
The role of the LED is to light up. The resistor is current-limiting -- but, more importantly, it builds a tolerance into the system. Wiring is designed to be low resistance, but none is zero resistance. This is especially true of aging connections that may exist along the way. The voltage drop is equal to current* resistance (i=v*r ==> v = i*r). So, under load, the resistance of the neutral wire matters more in the sense that it causes more voltage drop on that wire than on the ground wire, which isn't carrying any current. Now we have a potential difference across the resistor-LED path, which drives current through that path, potentially saturating the LED and turning it on.
The same thing can happen if the neutral line becomes high-resistance for some reason unrelated to load. For example, a bad connection between the shore box and the shore-side plug, or a bad connection between the shore power inlet on the boat and the inlet side plug, or a bad connection at the back of the shore power inlet or where the neutral line connects to the main panel or ELCI, etc. This often manifests itself with other symptoms, e.g. lower AC voltage on the meter, or fluxuation as the bad connection is stressed by vessel movement, etc.
In older systems where, instead of an LED, there is a light bulb, either of these situations most often shows up as a less dimly lit bulb than a true reverse polarity situation. You can sort of tell it isn't the same "bright light" that exists with a true reverse polarity situation. But, in LED systems, some times the difference in brightness just isn't as noticeable.
Where am I going with this? If I had to take one blind guess, I would guess that the problem lies with your shore power cable, shore power inlet, or the wire between the inlet and the reverse polarity circuit. I would guess that current is leaking from hot to neutral.
If I had to lay a more detailed guess, I would guess that corrosion, debris, salt water, or dirt and water exist on the plug that goes into your shore power inlet or the inside of your shore power inlet that were serving as a path for leak current. I'd clean them out, dry them off really well, check the back of the inlet for moisture/corrosion/grime and clean it as necessary, and then see what happens. In your situation, not being able to see things, I wildly speculate that there is a good chance that this will fix it.
If it didn't turn out to be a shore-side hot-neutral leak, and I got a second guess, I'd guess that it was a high-resistance neutral. I'd check and clean all of off the connections along the way, paying particular attention to trying to find corroded connections, brittle broken wires, things pulled out of crimps, etc.
If that still didn't turn out to be it, and I got a third guess, I'd start inquiring about the load at the time and how it effected the situation, e.g. air conditioners, etc. If the load on the system was particularly high, my next guess might be that the load was too high for the wiring and that the natural resistance of the neutral wire was overcoming the resistance of the reverse polarity circuit, causing that LED to light up.
Wild guesses from the peanut gallery.
-Greg
