How to raise sinking deck post
I get the issue with the nails.  I think you need to try to take them out.
I would start with a sledge hammer and try to tap one of the sides inward.   When demo'ing a deck you are taking full swings to do the same thing but I would go really light.   All you need is a slight gap.   Once you get a tiny gap get a crow bar in their and try to get it popped out maybe to a 1/2&quot; - I would not go further.
This only works if you can work it out without the rest of the deck moving and getting angles jacked up.   So either you need to tap softer or just back out of this or try the other side and start over.
Once you get around a 1/2&quot; (doesn't have to be that far) cut the nails with a reciprocating saw.
I would support this corner while you do this by either jack it up or throwing up something like a 2x6 which should be plenty with no load on deck.  Now follow the same procedures for the other side and this side should be easier.
From there you need to raise the jack or 2x6 to level.   Then drill a hole in each side for a lag bolt.   Get your bolts in at level and then try to get one more hole on each side for another lag bolt.
Option 2...   Dig post out really really good .   Raise it up with jack and framing.   Note that you must have this secure well to get the concrete raised (this could be easier if they didn't go very deep hence the sinking).   Pour a couple more bags in or go down to the appropriate depth for your region.   Let dry then take supports out.
Either of these options could be a 1 day deal.   If you are dismantling a lot of the deck to get to post top (remember you have to saw those nails out) then I would try option 2 first.   Digging this thing out in a muddy yard might take 30-40 mins and will give you an easy view of how hard it will be.   This would be my choice if deck looked nice minus the tilt.