Steve
On some keyboard to keyboard or keyboard to DAW connections, there can be MIDI echo loops between the two ends that literally swamp the connections with MIDI feedback streams. Go into the PSR-S770's MIDI setup menu and see if LOCAL is ON. If so, turn it OFF, and see if that resolves your problem. If you do this, and later want to use the S770 on its own, don't forget to turn LOCAL back ON, or you will not get any sound out of it. I suspect on your other equipment, LOCAL is already OFF, or by its very design, does not echo incoming MIDI data back to its source to begin with.
If the above is NOT the problem, then it would appear that something is generating errant NOTE OFF messages. While I can see maladjusted DAW software doing this, I can not think of anything on the S770 that would do this, and if the 61ES were at fault, you would see it with your other equipment.
Best of luck!
Roy