1) We can feel a bit of sway on the top deck when the kids are running around. Can I add some bracing/gussets to the posts to stiffen it up a bit?
2) The lower deck has posts on deck blocks that have been buried in the dirt and are rotting pretty badly. You might also notice the lack of railings. Would I be able to tie into the upper deck posts (front/rear plates, joist hangers...?) instead? Would this also serve to stiffen up the top deck?
Indeed, knee braces at the posts will help, but also, diagonal bracing between joists will also assist in removing the swaying you feel now.
Start the diagonal bracing at the house and run it on a 45 degree angle towards the outer rim joist. One line in each direction would be best.
Short of cutting off a body part, the worst that can happen in woodworking is manufacturing really nice looking kindling. --- Quoted from lenaitch