Not sure where to start?

If it is soft, it is ruined anyway. Tear is open and take a look. The most likely leak for a dishwasher would be underneath, so on the floor. Look under the sink, assuming the DW is next to sink. Check hoses and pipes, plumbing and drains. Finding a leak under the sink should be too hard. A few other thoughts depending on your setup, location of sink, DW, and other factors.

Go ahead and buy a new dishwasher then start ripping open you wall.

Make sure you find the source of the leak before buying anything. Could be from the water inlet to the refrigerator. I've seen those have a minuscule leak from where the homeowner had put one of those sharkbite connectors.

Also could be as simple as a hoseclamp improperly placed or not placed at all on the dishwasher, etc.

Summary: find the source

FYI - House built in the 60's. Had a leak from the kitchen sink/dishwasher drain. Couldn't find it there. Located vent stack on roof and eyeballed it down to wall vertically and then lined up the drain line horizontally. That was exactly where I could see moisture coming thru the brick on the outside wall, right after running the DW. Cut out that section of wall on the inside. (3/4" wood t&g panels, under the counter-top, real PITA to get to)
Found that sewer drain line was 2" thin wall copper and it has "split" at the bottom, probably from years of liquid drain-o, right at the stack connection.
There enough pipe left intact to cut out the split section and replace with rubber sleeves and PVC, since I don't know how to solder pipe. That wasn't a "professional fix" but will certainly last for several decades. No telling what a plumber would charge to fix that.