Welcome to the wonderful world of B-W O/D. To answer your questions, none of the O/D electrics care about polarity. The solenoid does, however, care about voltage. It's an electromagnet and its operation depends more on amps than volts. Since a 12v system has 1/2 the amps of a 6v system the solenoid, though it will operate on 12v, won't like it much and will get hot. If it's in good shape it can last a good long while on 12v but it would be best to replace it with a 12v unit or place a voltage reducer in the line between the relay and the solenoid. There are a couple of sellers on eBay that have come up with reducers that actually work for the solenoid; the generic ones you get at the auto parts store or the usual suspect old car parts suppliers won't work. Been there, done that.
You do have to lift off the gas to shift into O/D. Depending on load you may have to lift completely. On light acceleration just a slight lift will do the trick and sometimes the shift is so seamless you may not even notice it. Usually you will hear the solenoid click and feel a slight lurch when it shifts up. Most of Ford's O/Ds were 30% so you should see RPMs drop by about 1,000. If you have the O/D enabled (handle pushed in) and it freewheels when you lift off above the governor cut-in speed then the O/D is not engaged and you'll need to diagnose it. Most O/D problems are electrical.
Go here for the official B-W operation/maintenance/repair manual:
http://www.oldcarmanualproject.com/manuals/trans/Borg-Warner%20Overdrive/index.htmBTW, if you change the oil DO NOT use anything marked "hypoid". Many modern manual transmissions use it but the additives that make it hypoid will attack the brass synchronizers in vintage transmissions. The proper oil can be had at NAPA (if they don't have it they can get it), Tractor Supply (they call it Ford tractor transmission oil), and probably CO-OP and other agricultural suppliers.