Thanks Dave.
I looked back at my notes (I keep a logbook, yea, dork, I know) and the first mention of the noise was 350 miles ago. At that time I thought it might have been related to taking a sharp turn. With that in mind, I thought it might have been the metal portion of the coolant pipe touching the bottom of my metal coolant reservoir. I put a piece of foam padding at the point where they might have touched, but the noise continued. So while I've heard it while turning, it's not as consistent as it is with accelerating.
The gear whine mention under acceleration or deceleration is the closest match to what I am hearing. I cannot get the noise to repeat if not actively pressing the gas pedal. The car can be in motion at any speed, but there's no noise if you aren't pressing the gas pedal (i.e. coasting while not pressing the gas). Further to that, it is much more evident when initially pressing on the gas and also at times when letting off the gas. However, when keeping a reasonably steady push on the gas pedal, the noise will fade away and stop.
So, yes, much more consistent with accelerating and decelerating than anything else. Almost to the point where I can't get it to repeat doing anything else. I'm not sure what a gear whine might sound like exactly compared to my own noise I'm hearing, but your description of under accelerating (and to a lessor extent decelerating) sure does fit.
For the motor and trans mounts, this could also be it as it's related to accelerating. My mounts are not in great shape. The motor mounts are rusty (like other brackets and fittings on my car have been before replacing them) and the rubber is of questionable condition. The trans mounts are less visible to see frequently, but looked more rusty than the motor ones on my car. I will definitely look closer for evidence of the pivoting like you described. I have not "felt" the noise though, not the seat of the pants variety anyway. It might be vibrating enough to feel, but it's not as obvious as the noise, so either I missed it or it's faint or not happening. Possible though still.
I would not think it is an engine miss. The engine seems powerful and okay when giving it gas. I had the shift computer rebuilt years ago and back then it had done the thing where it stayed in 1st gear and rev'ed at high RPM, but nothing like that for years since the upgrade. About the only sluggish feel I get is sometimes when first leaving my house going for a drive and it wants to hesitate slightly before changing into 2nd. That could be me not pressing the gas pedal down as much as I think I am or a cold engine and trans needing some warm-up time to get the fluids flowing (whatever that might mean). Hearing others describe missing engines doesn't sound like something my car is doing. I'd rate that as unlikely for the time being.
If it is a gear whine that isn't improved by adding differential fluid back in, what would have been damaged and what do you eventually need to do to correct it?