Am I understanding you correctly, the car is locked and will not budge at all backward even if pushed? If so, can you push if forward or will it drive forward?
If it is totally locked going in reverse, and the solenoid is installed correctly have a look at service counselor vol 21 #15
https://www.packardinfo.com/xoops/html/downloads/SC/SC-VOL21NO15.pdf and see if any of those symptoms fit. Even though it's a later bulletin, the overdrive is virtually the same as what was used on your car. Perhaps when replacing the unit, something was misaligned or is now binding to the point the sleeve mentioned as the problem for driving in reverse is locking things up.
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