Problem #1:
tippy one way due to the foundation not being done properly
It's tippy due to the way its designed and loaded. No matter how you do the foundation, this isn't going to change. You either need to get rid of the top rack or put in cross bracing on the back side. Your design is inventive, but different than every other firewood rack out there that you would buy commercially. If you have the full stack sitting on the bottom rack, it won't tip, but the moment you load that top shelf, the whole thing is going to rack without cross bracing. In addition to your aesthetic concerns, it's probably dangerous. Depending on how wood is taken off and removed, the whole thing could topple. (leaning or not)
You can keep some bracing at the top, but I would keep a single stack of wood top-to-bottom in the rack. Eliminate the shelf.
How would you recommend establishing a solid foundation below each of the legs to correct the sinking and tipping?
Footers are a really expensive way to go. You can have the whole thing float on the ground and heave with the seasons. (Its just firewood.) The key is spreading out the pressure. Instead of having all the weight go through 6X6 patches, put the posts on pavers or larger blocks of wood to spread out the pressure. Add a 6" bed of compacted gravel underneath that to further spread the pressure to the ground. Add some little legs in the middle of the bays to take even more pressure off.