"E-brake no function" is, at least around here in Ontario, pretty much a guaranteed "out-of-service" violation in a roadside safety check. Especially if it wasn't hooked up to begin with.
However, it shoudln't affect the adjustment of your brakes. As PowerMad points out, backing up and applying the brakes is
supposed to do that through the self-adjuster mechanism. Those get seized up too (both the parking brake cables and the self adjusters seem to have been designed in a non-road-salt-use part of the country

) and quit working, which may be the case in your situation.
You can try adjusting the rear brakes with a brake spoon, through the slot provided in the backing plate. If that doesn't work, you may be into pulling the drums again and replacing the adjusters. At the same time, throw in new parking brake cables.