Can completely shot motor mounts and the resulting engine movement cause shifting problems?

The car I'm driving has completely shot motor mounts. The engine jumps quite a bit, especially when you put the car into gear or take off (1st gear) from a stand still. Therefore the attached torque converter and transmission also have movement. The car sometimes doesn't seem to want to leave first gear and the RPM's get up there. Even letting off doesn't necessarily get it to shift. However, the car has only done this a few times, not all the time. In all other gears, it shifts well and smoothly, especially for being as old and worn out as it is - it's a 1997 Ford Taurus G. I also have noticed that it's not a cold problem. I drove the car and paid attention while it was not warmed up and it shifted just fine. I'd also like to say the transmission fluid is full and seems fine, although I'm considering have a drain and refill. Anyway, and with all that said, my question is if the broken motor mounts and movement of the engine/torque converter/transmission it causes might cause the problem I've described. I hope it's not actually the transmission, but I'm just not sure. What do you think?

Added (1). Texas- It's funny that you have a condescending tone as though you know what you're talking about when it's apparent from what you've written you don't know much at all. The motor mounts are broken. That's why the engine rocks. There's no question. Broken motor mounts are a clear problem. And there's no welding to correct hydraulic mounts *rolls eyes*. I will be purchasing new mounts. They don't cost much. As far as theorizing about the transmission, that's all I can do at this point.

Added (2). *Update*

I would like to disclose here for anybody who checks back or any future people who come across this question that I've discovered my car has a second and probably completely unrelated problem that is causing the symptoms I've described - a vehicle speed sensor that needs replaced. In addition to the symptoms I described, an erratic and sometimes non-functioning speedometer is the other symptom I've discovered that ties into all the others and strongly suggests the vehicle speed sensor.

That certainly is possible, you or someone need to check the mounts from under the car. It does sound like you have 1 or more broken motor (and maybe tranny) mounts.

If engine and transmission are moving around operation linkages change lengths and positions. This makes shift location not where they are expected to be. Also if throttle is a linkage, if engine moved throttle could go to full on.
Definitely unsafe.

Being a FWD V6 it is quite necessary to have good motor mounts. The transmission is fastened to the motor, and so any movement affects the transmission in a direct way. Shifting problems may arise due to vibration (dislodging internal/external components). Please have a good transmission shop take a look at it. Are there any transmission codes popping up?

Rather than strictly theorize, get under the car and look at the motor mounts, or if you aren't comfortable doing that, find a shop that can check it for you. If you inspect what is wrong, you may either find and pull motor mounts from a similar junkyard taurus, or weld/bend/repair the existing ones if it looks worth doing. Anyway, if you just theorize and decide to ignore it, you may find the engine and transmission just falls out on the ground someday.