Certainly. You will need a sink strainer, and a lot of new piping under the sink. Not as imposing a job as it may seem. At a hardware store, a strainer can be had for under $10, and a plastic end outlet waste kit for another $10 will give you most of the parts you need.
The dishwasher will have stale residual water in it causing an odor. I would pour a gallon of white vinegar into the machine, then take the drain hose and lay it down completely flat so all the water and vinegar will drain out into a shallow pan. If you feel like turning off the electricity to the DW, you could remove the lower panel and disconnect the drain hose right at the pump.
Possibly spritz the entire inside of the machine with white vinegar. An abandoned DW tends to be a source of odors.