Before you remove the overdrive from the car check three things:
1. Check the oil level. It should be even with the bottom of the fill hole. The lack of sufficient oil will affect the shifting. Borg Warner overdrives are notorious for leaking oil through the rear seal. The oil will not drip on the ground, it will pile up in the banjo of the rear axle assembly. Pull the fill hole on the banjo and see if it has manufactured additional oil.
2. Check that the solinoid is working, you should be able to hear an audible "click" when power is applied. make sure it is installed correctly if it has recently been removed. Remove the two mounting bolts and pull straight back on the solenoid. It should move against the spring load, but not come out. To remove it you have to rotate it 90 degrees. It should be installed the same way.
3. Check the pull cable. It should be able to move the shift lever to the full forward and aft stops. The cable may have slipped where the end is attached. If the lever is not in the full stop position it will not go into overdrive.
If none of these options work you will have to remove it from the car and disassemble it. The only person I know that works on them in Northern California is Bill Swigart in Redding, CA. 530-221-1628. He has a cottage industry modifying BW's for Model A's. Howevwer I think he may have retired from the business.
Tom Endy