Do you know what rpm 60mph is? Does it vibrate at that rpm in neutral? If it does not that may eliminate the engine, trans, and clutch.

Is this a new problem or did it show up after some repair was done?

You said you inspected the u joints. Did you inspect the entire drive-line for evidence of missing weights? Maybe a weight fell off.

Will the place you purchased your tires re-balance them?

When was the last time the car was tuned? Could it be missing in a high load condition. Maybe a bad plug or wire.