It sounds like your park switch (located inside the gear box of the motor) is messed up. My wiper motor didn't work when I bought my 68, so, wanting to keep all original, I followed the service manual and drilled out the crimps on the gear box and repaired a broken wire. While in there I noticed a cam and copper/berrylium contact that stops at the same position everytime power is removed from the motor. I can't recall all the details as this was 2 years ago. The service manual has a pretty good description and pics. If you want to get that involved, go for it. Otherwise you will need to get another motor.
Do the wipers stop at the same place everytime? Do they stop at the top of their travel? If so, you need to rotate the drive arm on the motor.
Hope this helps, as I am far from being expert in this matter.