I was recently in the same situation - an older deck with 24" on center joists that I wanted to replace with composites. In removing the original decking I discovered that the original joists weren't in very good shape - some rotting here and there, bowing, etc. I ended up replacing everything except the posts set in concrete and 3 main beams, which I used as supports by attaching new 2x12's to the old beams, and rebuilding from there. By overbuilding at each step, I think I have a very solid deck, with new composite decking.
Now my deck should last a couple more decades and it looks great.
As the previous poster said - Do it right and do it once.