Adding to the intrigue... I decided to just leave the projector alone when it started acting up. I quickly discovered a pattern to the behavior of my projector...
When it was acting up, For EXACTLY three minutes, the picture would be perfectly fine.
After that, the picture would begin to shake and eventually diverge into two identical, yet narrower images. Then the two separate images would begin to shake and eventually merge. I bet you couldn't guess how long it took to go through the shaking-split-shaking and back to normal stage.... EXACTLY three minutes.
Shaking image - about to split
Starting to split
Split image - shaking and about to merge
I timed through five cycles... each time, it was exactly three minutes of a good picture, followed by three minutes of a flawed picture.
At this point, I'm wondering if my Extron box is malfunctioning. It is SUPPOSED to detect that I have only one Sync cable connected to my projector and output a composite sync.
Is it worth picking up a BNC Y connector and hooking both the H Sync and V Sync from the Extron box to the composite sync port on my projector? I know this isn't a good long term solution as it would risk frying my video card... but would it hurt to hook both up temporarily to test?