Exactly the same problem (and more) for me, and it started as soon as I updated to iOS 16 two weeks ago. Prior to that I was on iOS 15 with no issues. I've even had podcasts auto-delete as I was playing them - initially I'm playing the downloaded version, and "Remove download" is an option, and within a minute or two I'm apparently streaming the podcast because that "Remove" option has been replaced with "Download podcast". I select that, and it downloads, then the next time I look it's once again been deleted even though it the podcast currently playing.
I definitely do not have storage optimization enabled - I prefer to handle all that myself - and it's a 256GB iPhone with 176GB available. And "Remove Played Downloads" has never been enabled, which shouldn't be affecting podcasts that have never been played, which are magically auto-deleting.
Possibly related: I first noticed something was wonky because I've always downloaded podcasts to my Mac and then synced them via USB, but as soon as I updated the phone to iOS 16 some of my podcasts refused to sync, even though the checkbox has been ticked to sync. Some do sync, but some simply don't without any kind of error message - they just don't show up on the phone (or maybe they do sync and are instantly auto-deleted; hard to know). I tried a number of things to fix this behavior - unsyncing and resyncing the particular podcasts, same thing with all podcasts of a particular show, restarting both the Mac and iPhone - all to no avail. At that point I gave up and simply downloaded directly to my iPhone, and found that once downloaded the same podcasts will auto-delete within a few minutes, even though they're marked "Saved" (both on the Mac and on the iPhone).
I looked into my data usage with my carrier, and even though this only started two weeks ago, I've already used more cellular data than I normally do by this time of the month, presumably because podcasts are streaming while I'm out and about rather than playing from copies downloaded over WiFi. So it's not just annoying, it may wind up costing money.