OK - having just posted a response, I went back and did some digging.
First, this is a known property of the run/stop jack on the DFAM. It was called out in the Sound on Sound review a while back:
https://www.soundonsound.com/reviews/moog-dfam
I am surprised this hasn’t been fixed, as it is an unconventional (and not very useful) feature.
That said, I found the following workaround, which requires an external VCA:
(This assumes that you are using an external clock to run the DFAM via the ADV/Clock jack.)
Clone the clock signal that is going into the Adv/Clock jack and send it to the VCA. Gate the VCA with the same 0-5V gate that you are feeding into to Run/Stop jack. Take that gated clock signal and send it to the Trigger input on the DFAM. This “gated clock” is very useful, as noted below. The input to the trigger jack triggers the envelope on the DFAM. Because it is gated by the same signal that is starting/stopping the DFAM, the clock will only be present when the gate is high.
Of course, there are other variations possible:
If you don’t gate the clock signal through a VCA, you will retrigger the DFAM at which ever position it stops on (until the Run/Stop input goes high again).
Alternatively, you can skip the input to Start/Stop and simply feed the gated clock into the ADV/Clock jack. This is almost the same as using Start/Stop, although it may be less precise (not sure about this).
Third, you can use the gated clock but not feed the gate to the Start/Stop. In this case, the DFAM sequencer will continue to advance, but the sound will only be triggered when the gated clock is on. This is useful if you want to keep the sequencer synchronized to some other part of the patch, but you want it to fire only periodically.
Finally, and most simply, you can feed the DFAM through an external VCA to gate it on and off, while allowing the sequencer to run. This is essentially the same as the last case.
I hope this helps!