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.

Interesting.  My 55 Bird has O/D.  Before I understood O/D, I drove my car around with the level pulled out.  This is mechanical of course.  I would think the solenoid will still work, but the pawl is held out keeping the O/D from engaging.  Sounds like you may have a wiring or relay issue.  Does the O/D work with the lever pushed in?

The lever doesn't control the electrical circuit it simply "locks out" the overdrive. 
The governor acted up on one of my overdrive cars and as a result the transmission stayed in overdrive . When stuck in overdrive you can't back up without doing damage to the overdrive unit. I changed the governor and also installed a toggle switch under the dash to control the power to the solenoid.
I leave in the on position when I want overdrive function and turn it off when I don't , that would cure your situation and it also provides a way to get the unit out of overdrive if you have a governor issue like I had.

Pete

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...

The OD cable simply enables (pushed in) or disables (pulled out) the OD. The solenoid is energized at 27 mph when the governor contacts close and ground the relay and is de-energized at about 21 mph, both regardless of cable position. The solenoid contains 2 coils; the high amperage pull-in coil should disengage the instant the plunger moves toward the drive plate regardless of whether or not the OD is enabled. If it doesn't disengage the points are out of adjustment. Adjusting the points requires removing the solenoid and the cap.. You can observe point movement by applying battery voltage or just pulling the plunger out by hand. Adjust the points as needed. A toggle switch will prevent solenoid operation when you're driving with the OD locked out, but if the pull-in coil points aren't disengaging the solenoid will get hot because it's drawing twice the amps it needs to hold the plunger in the engaged position. This will eventually burn out the windings and you'll need a new solenoid. Ask me how I know...