When I’m driving the car above 27mph and the od lever is out, not engages the solenoid is drawing a lot of power trying to push the pawl. How does moving the od lever in and out shut off the electrical to the solenoid? When I drive around town I don’t turn on the od. Solenoid gets hot because it never extends and never gets to hold mode.

According to the diagram in the '57 Ford shop manual, that's the way it is.  I never realized that.  The Borg Warner overdrive manual shows the same thing, except some models had a switch that interrupted the governor circuit.  I once had a Lincoln OD that had that switch, but I didn't know at the time what it was for.  The switch was actuated by the end of the shift rail that was actuated by the manual lockout lever or the reverse lever.  A toggle switch in the governor circuit would do the same thing.  I just realized, if you exceed 27 mph in reverse, the OD would engage and destroy itself internally with the stock wiring setup.  Wonder if anyone ever did that.

John - "The Hoosier Hurricane"

Burt Reynolds...