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My left side was “jiggley” when I got the car and upon investigation found the original bolt/pin that connects the lift arm to the assembly had been replaced and the fit was loose.
I suspect that if both lights are moving around some it could be that the plastic bushings that the lift arm pin goes through could be worn or brittle to the point they have fallen out.
I lucked out by finding a member in the classifieds parting out a headlight assembly that had just what I needed. Putting the correct pin and bushing back in place corrected my issue and the lights have been rock solid since.
My immediate guess would be to check the bump stops mentioned above. The bump stop rubber isolaters - you have four of them to each headlight, two that control the full open and close and two that "hold" it in place when open - deform with continuous use. As a temporary fix, you can remove each one and flip it 180 degrees so the "good" side contacts.
You'll see the two for the headlight door pretty easy. When you open your hood they're immediately on the "top" side of the headlight near the harness and motor ****. The two that control the full open and full close are a little harder to see. Raising the headlight motor manually via the ****, they will come into view on the headlight motor bracket. Obviously you can't raise the **** all the way otherwise you won't be able to actually remove the one that controls the full open position.
Other than those the only other items to check would be the physical mounts of the headlight assembly itself, which has a possibility of being loose, and also the nut that holds the pivot arm to the headlight body.