If you are getting 12.35 immediately when disconnected and the 12 volt accessories not working then you most likely have a circuit breaker or disconnect switch not working.
With the trailer still plugged in check the battery and see if you are getting in the 13 volt range.
If not go check the fuses in the trailer. If they have the same reading as the battery then your charger is not working. However if the lights come on on shore power and the fuses read in the 13 volt range then you have a problem with power getting to the battery.
