Cover/wrap 6×6 post with plastic to protect them from rain during construction

If I was making a shim, I would not want the grain running the short way since it will tend to disintegrate under load and time. Ideally 7/8 plywood, partly because it is harder wood... otherwise plane or rip 2x6 or two 2x4s to the desired thickness, or 1x6 if the thickness will suffice. I would also glue and screw it.

Check the tag on the end of your posts, see if it says it is rated for ground contact. If so, you're fine. The post base simply keeps it out of standing water. If it says above ground, you are still probably fine. That post will last longer than you'll enjoy your deck. Honestly, that worthless little post base will rust away before the post rots. I've repaired that issue far more than I've repaired rotted posts. Modern engineering does not concern itself with longevity.

However, if you feel compelled to address the issue, use a heavy-rated automotive jack rated for 5-10 tons+. You can very easily jack up each post a few inches to do whatever you feel you have to do. Don't bother with the commonly-available floor jack you have in your garage, they are typically only rated for 2 tons, and will fail.

My advice is fill around each post with larger-sized gravel for drainage, backfill as necessary, and forget it.