Don't trash them! The growing parts are taking food and water from the tuber; which is why they are shrinking. They need to be planted soon. You can cut them into pieces with a growing 'eye' on each part. Once in the ground;, the tuber itself has done it's job and and will rot.

Thanks for the reply, rain1950. Much appreciated.
I'm excited to see what happnens with my first try at potatoes.....

I actually had a similar problem. I accidentally left two bags of seed potatoes from Walmart in the car for about a week before planting. When I took them out, they were sprouting like crazy. Luckily, they were all pretty small and required no cutting so I just aimed the sprouts somewhat upward and planted them. I tossed the ones that looked like they were truly rotting (black with white mold) out into our old garden area. I had plenty left to plant without those so I didn't lose anything.

These particular potatoes were an early variety and are already 6-8" tall after only a couple of weeks. The later variety that I planted (which weren't already sprouting) a couple of weeks earlier are only about 2" high. If I get a decent harvest from these, I'll be leaving my seed potatoes in the car next season, too!