Replacing Rotted Deck Footings
You can dig out the old footing where accessible even if means digging an "offset hole" to get it out. place a sonotube back in the hole, 6" above the bottom and fill the tube either 2" above grade for a short post or mount the clip to the deck after the old post is removed and run the concrete up to the clip. Place a wood form in the bottom of the hole if the offset is too much. 

To get access to the other holes, remove 4  to 6 decking boards the get access from above, and do more of the same there.

None of it is going to be easy, but it is possible without tearing out the whole deck