Meadowlark,
What is happening, is your pier is "racking." It's a term used in home building that also applies to patio decks and in yout situation, piers. In houses, the best way to stiffen up the wood framing is sheething. OSB is the most common material and it's very effective.
On wooden decks, the house usually holds the deck solid, or the posts are too short to bend on you. If you build a deck tall enough and far enough away from the house, it will wobble too.
In your situation, your posts are tall enough to bend and flex. To stop this, you need diagnal bracing from the bottom of one post, to the top of the next one. This creates a triangle, and it will eliminate the the racking. Imagine an old wooden train trestle, and you'll get the idea. The bracing will tie the top of the next post into the bottom so that it cannot move or flex. With really tall posts, you need to tie them in at different heights, depending on the height of the post for maximum integrity, but for what you are doing, there's no need for a real engineer.
I'd also do every other set of post side to side in the same manner. Maybe go in oposite directions to change it up. You can also creat an X, if you get in the mood, it won't hurt to do a few X cross braces, but you don't need to.
I copied your picture and drew some quick lines to give you a rough idea. You don't need to attach them at the very bottom either. The water line should be fine.
They also don't need to be very thick or heavy wood. Pt 1x6 will work great, or PT 5/4 decking will work too. I've used both. Just be sure to use good screws rated for PT wood, or galvanized lag bolts.
Hope this helps and I was clear.
Eddie