Hello Phil,
Some years ago I had a TR-707 in my workshop with pretty much the same issue as yours.
Just like Florian Anwander suggests, it was easy with my scope to trace the source of the issue: It turned out to be a burnt out TL064 in the Envelope Generator. A new TL064 and the TR-707 was like new again.
Just for a precaution I actually changed both TL064s. It seems like these TL064 JFET opamps are prone to fail in this manner.
A couple of TL064s won't cost many $$, so Phil, why not take the chance and hope that your TR-707 has exactly the same as the one I repaired
I wish you good luck. Keep us posted on your progress
Cheers
Bo