Shift cable and the mechanism?

Those are a push pull system. When you move the shifter lever inside to the right and left the cable on the transmission end pulls the internal shift fork selector up and down with the shift selector shaft linkage. Then when you move the lever front or back, it rotates the top lever outside the transmission. The internal selector then moves and slides the correct shift yoke in the transmission to select the gear you wanted. So say you wanted 1st gear. You move the stick inside and the cable will move the internal selector into the notch on the first gear yoke. Then when you put the stick into 1st gear it slides the yoke and moves the sliding gear in the transmission. There are some adjustments but the common issue is the bushings getting worn and the extra play from that means the transmission doesn't fully engage. That is when you replace the bushings or the entire cable assemblies.
What is the issue you are having?

I was looking under the hood and the part on the transmission the cables are hooked up to is oily and the rubber on the cable is dryrotted

That is likely from seepage as the shafts move a lot and can carry a light film out of the seals as they move. Wouldn't be hard to use some cleaner and remove the oil, but it won't really hurt anything. The rubber is there to keep moisture out. The easiest solution to it would be to use some liquid tape are brush able rubber (like plastidip or flex seal) and coat them with a layer. They do sell replacement cables but if yours are working OK just coating them would work, Or wrap them with harness tape and give that a coat of rubber.