Thanks norm123. I tried again and was able to delete SOME, but not others, so that helped a little.
There seems no pattern in terms of date, either. Four episodes download each day. It also doesn't matter which podcast I'm working in, a free one from iTunes or one that I have paid for from a third party.
There are about 15 individual episodes I've tried to delete without success, even if I download and listen to them (or fastforward) again and then try to delete. Other episodes in the same podcast series downloaded a day or before or after will delete without problems.
I've restarted iTunes and rebooted my computer. When I highlight one of the problem episodes, the Delete key is nonfunctional, and in the menu bar, the Delete option under Edit is grayed out.
I also tried all of the optional settings. I haven't tried "just save the last 10" because there are about 100 and I don't want to lose them (a long trip coming up, gotta have fresh podcasts to keep me company). Even when I choose "keep all unplayed" the played ones don't go away.
I have switched to the "My Podcasts" view and it doesn't show the zombie podcasts, but it has them in the wrong order and is hard to manage because of the size and spacing.
P.S. I asked a similar question on another thread Titled "deleting old podcasts", and Roger Wilmut1 said "this is a recent change to the design of iTunes - I think it's done to bring podcasts into line with songs in the way it operates, but it's annoying a lot of people. At the moment there's nothing you can do about it, though the "My Podcasts" view shows only downloaded podcasts"
At least I was able to get rid of SOME of them after trying again, as you suggested, but I look forward to the time when it is consistent.