Toyota Repair: 2002 Toyota Avalon XLS Severe Vibrations, toyota avalon, steering column


Question
Sometimes when driving my 2002 Avalon XLS, the car begins to shake/vibrate very severly.  It feels like it's the steering column and I can feel it in the brakes, too.  If I pull over and wait awhile and then drive again, it will be fine.  Sometimes when I can't pull over, it will eventually stop vibrating.  The dealer thought it was the tires, so I bought 4 new tires and thought the issue was resolved.  Unfortunately, it is not.  Of course, when I bring it to the mechanic, it doesn't do it.  The shaking is very severe and there is definitiely something wrong. Help, please!

Answer
Is the shaking there without applying the brakes or only when the brakes are applied?
Besides having the tires replaced have any other repairs been done?
This all sounds kind of strange because an imbalance condition in the drivetrain will not be intermittent it will be there all the time.
To your knowledge has the car be involved in an accident?
How many miles are on the odometer?