Draining the battery? Well, shucks, I didn't know that Airstream wiring was made by Lucas, Prince of Darkness! (For those not in the know, Lucas did the wiring for many early British sports cars, which are well known for their electrical problems.)
Ok, so here's the way to locate the drain from my experience with the MGB. Get yourself a wiring diagram and a little ammeter. Turn everything off and test each circuit. If you've got a short someplace, that circuit will show something on the meter. Keep tracking until you locate the culprit.
Of course, if there is no drain, then you've just got a crummy battery.