I think there is something in code about a deck post needing to be same depth as house footing if within 5' of the house
But I thought that was only if the deck is attached to the house
So the problem with using the old posts is that the new posts would need to be dug in
I'm not sure of the soil in your area, IE frost heave
Seems like the best bet would be the blocks that you set on the ground & then set the beams on top of the blocks
OR there would be a short post to bring the beam up to where you want it
Example, but best solution is the beam sits right in the pocket of the support
The newer blocks have pockets for the beams to sit in, or a post as shown