Some background history is always helpful - when did this start, anything notable prior to this behaviour? (even if you do not think consequential)
Do you have any non-OEM accessories installed? If so what and to where are they connected?
Just to be sure I'm understanding correctly:
your EMS remains on even with the Ignition OFF, yes?
Have you tried swapping the EMS and say, the Fan Relay?
(you would get instant verification on relay being 'stuck' if the fan runs immediately)
If swapping the relay makes no difference check the relay socket for a short between the 30 & 86 terminals.
Does it make any difference to relay EMS being on if the Kill/Run switch is in the Kill position?[/QUOT