The first thing to do is to fix whatever leaks you have, this can be a small leak so check thoroughly.
Cabinetry is basically box building. The floor or bottom is part of the structure that holds everything together. This is put in place by the use of dado cuts otherwise called grooves. You will need to remove the bad material completely. If you are able to remove the material in the groove then you can install nailer strips into the grooves using wood glue and screws. If you can’t remove the material in the grooves. Then place the nailers at the joint line and attach with glue and screws.
Then you can insert a plywood bottom in place and ***** down on the nailers.
NOTE this sounds easy but will require some Patience cutting everything to fit drain pipe, etc.

The dampness you feel means that something is still leaking. Fix that leak before you do anything else, otherwise, the new stuff will rot too.

Really funny you should ask this. I just spent my weekend doing just that. First I bought a new kitchen faucet. Then I bought some mold/termite/wood rot spray and sprayed the shelves and wood underneath the sink. Then I covered the wood-treated shelves with new contact paper. Then I caulked around the edge of the sink. The damage you refer to is called wood rot. It comes in many varieties and forms of mold. Any good hardware store will sell you a spray-on solution to treat the wood. It is important that you do, because it’s only going to get worse. Happy home improvement!