Could Broken Engine Mounts Cause Shaking And Shuddering Between 60 And 70 Miles Per Hour

My car shakes and shudders on the freeway starting at around 60 mph, and ending around 70 mph. It started out gradually, a few weeks ago, but got worse very rapidly. When I ran a search on that symptom on car related websites, the diagnosis that always came back had to do with the axle assembly or suspension. THe most common suggestions were to check the ball joints, cv joints, and cv boots. So is the cause of this problem likely to be the engine mounts, like my mechanic is telling me? If it’s not likely to solve the problem, does it even matter? If I don’t replace a defective engine mount, is it going to damage my engine over time, so that I should really be getting it fixed right now either way? Or should I take my car to a second shop and get a second opinion? It’s sounding like the engine mounts will be a somewhat expensive repair. I have a 1996 Toyota Camry with approximately 125,000 miles on it, if that matters at all. Thanks a lot.
Sorry everyone. I should have included in my original post that I have brand new tires, and I’ve already checked the balance. So it’s not the tires.

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