you can get under the deck correct? Have you crawled under the low area to eyeball the situation?
Is the deck structure a set of beams with joists spanning the beams then the decking is fastened to the joists? Are the beams fastened to the posts with carriage or lag bolts?
My thinking is, if they are, then you could rent a floor jack, maybe two and set the jacks on cement bags and then loosen the bolts and then with the aid of a level, jack the beam up in unison until full bubble or level. Then with the use of Dek Blocks and 4x4's make new posts to bolt to the beams and this should do it. You would not have to dig holes, because you physically cannot, the Dek Blocks would take the place of the holes and do the job. I would use them 2 feet apart and make sure that the ground under the Dek blocks are hard and sound.
What do you think and what will make this not work on your situation?