There was a recall for cars with electric steering a few years ago. Do you know if your car had it done?
The mounts hold your engine firmly in place, so that could be the cause of your vibration. As you hit higher speeds, the engine's using more power so the vibration would grow. Here's a not so great video, but it might give you an idea.
I haven't done motor mounts before, but here's a sort of okay video on how to do them. The guy's a bit of a goof, but the video's aren't bad.
Depending on the oil change place, they might be exaggerating the shocks and struts. Usually places that are $30 for an oil change up sell like crazy to make their money. If you're driving and you hit a bump and your car bounces like crazy, your shocks/struts might be gone. You can also push down on the four corners of the car and if it bounces a lot then they're bad.
If you end up wanting to change them, here's a few how to videos. You could do them yourself since it's more tedious than anything, especially if they're stuck in there good.
My G6 was also in an accident in 2012 when it was hit on its side. It took about six months of pressuring the shop and the insurance company since they just kept giving it alignments and saying I kept running over curbs, etc.. to ruin the alignment. When they finally dug into it, they found the impact had damaged the steering column, steering rack and steering module. They had to replace all three, but once they did it worked perfectly. However, if that had been your car you likely would've noticed it immediately after the accident. It's still a possibility your car could have a similar issue. I think the broken motor mounts are the likely culprit.
Unfortunately, once you get in an accident it can start jarring things loose or accelerate things breaking down on the car.