I was hoping that Apple would have updated iTunes to fix this issue. I started having this same problem, but not just with podcasts. I realized it was probably iTunes when I had problems with my music and movies. My music play list would scroll through until it would find one it could play. Movies just wouldn't play even though they were there. I tried to dump iTunes in the trash, but under Maverick you can not delete it. In the old days, pre OS10, a corrupted preference file used to cause these kind of problems and re-installing the app never replaced the preference file so it never fixed the problem. With that in mind I went looking for a preference file, but could not find one. I did delete some file in the music folder that temporally fixed the problem, but I also had to rebuild all my play lists.
I would also note that no mater how I set iTunes, ie not to sync any podcasts that when I would sync my iPhone it would reset the podcasts showing that I had not played them yet. I've had sync issues ever since I upgraded to iOS 7 and was forced to use the podcast app. Although I still think the problems are with iTunes itself.