Disconnect the battery and leave the TT plugged in to shore power. Make sure the battery cable does not touch anything if you disconnect it. Mark your battery cables POSITIVE (99.99% of the time it is the BLACK cable and the NEGATIVE cable is WHITE (The negative cable (WHITE) should go directly to the TT frame. The POSITIVE (Black) cable should have a 30Amp inline fuse near the battery.
The reason you need to disconnect the battery, is because if it is a DEAD battery and it is not going to hold a charge, the controller is probably having too much of a load and is dropping off and retrying. Being 8 years old, it probably does not have a smart battery charge controller so it will continue to try and charge a dead battery... boiling it.
Disconnect the battery and run your shore power to see if the light problem goes away.
You need to let us know what the battery voltage is.
Don
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RVing with SOLAR