"When I apply the brake I have a moderate shaking that I have never noticed before. I don't think the tires need balancing or the wheels aligned. What other parts could cause a shimmy only while braking?"
In my experience it can be only one thing!
The exact same symptoms occured to my Mystique. It was shortly after a front rotor and pad replacement and yes a tie rod was loose and replaced, still the vibration persisted. I checked the wheel nuts for torque and found they were unevenly torqued and way too tight!
Re-torqued the wheel nuts to 75 foot pounds and changed nothing else. Result was that the vibration was completely gone after a week or so! You may want to clean the wheel's stud holes and ever so lightly lube them so that even resistance is experienced by each wheel nut. DO NOT lube the wheel stud threads as over-torquing will the result. Our cars have brake rotors that are very sensitive to uneven and/or over-torqued wheel nuts.
Give it a try, you have nothing to lose as the only other cure is new rotors torqued properly or the problem comes back again and again!
Regards,
John
