This appears to be a problem of "poor design" combined with "cheaply made cabinets that fall apart when wet" used around a sink that is installed to ensure they will get wet, due to poor design. Probably looked just like the picture in a magazine of a sink that was never connected to water for the photo shoot.
If you otherwise like the sink, install it correctly. That is, undermounted so that the countertops drain into it, not overmounted so that the countertops are guaranteed to get wet and stay wet. Looking at "farmhouse sink" images, I see three results that are not overmounted out of the first 32. If you happen to be wedded to this design faux pas, then you have to also be wedded to wiping up every single time you splash even a bit, and you need an elevated drip tray that can drain into the sink for drying your dishes. While re-doing the counters, consider also not using wood unless you are likewise committed to maintaining it in harsh service for wood countertops. Or, plane them down to solid wood, removing the rot, and finish with several coats of polyurethane to make them more water-resistant.
As for the cabinets, I'd probably start (being value-conscious) by seeing what I could find at the closest Habitat Re-Store for decent used cabinets. Or, think through your future kitchen remodel far enough to do the sink part now. These cabinets appear to be past redemption due to being made of a material that dies when wet, that got wet.