I have read this before. How are you able to tell if it is staying on? Never seen one in person do this.Check the rear wiper motor. Mine if used will stay running until it kills the battery. My expedition sits for 3 to 8 weeks at a time, to solve my issue, removed the motor to rear wiper and I added a 7.5 watt solar panel to the truck. Never had an issue. Even at minus 40c up here in Canada.
Battery. Excide 750 cca would be a great start.
I got an excide 1200 cca (cold cranking amp) in my expedition