For me, the following steps worked - sorta.
First, I didn't have "manage storage" at all when I plugged in my old iPod Shuffle today. (5/6/2020) It's been a few years since I used it, but I wanted to have something to listen to podcasts on at night that wasn't my phone.
In order to get the "manage storage" to show up, I had to click "restore ipod..." and reset my Shuffle completely.
(Note: I had the most recent version of the software already. (1.0.4) But it still didn't show "Manage storage...")
Second, I then clicked on "Manage Storage..." at the top.
This brought down an overlay that showed Songs, Podcasts, Audiobooks in the left nav.
I clicked on Podcasts.
I then opened the Podcasts app in another window and I navigated to the downloaded episodes.
I dragged the episodes one by one into the Shuffle overlay that was already open to Podcasts.
"Nothing" happened. This was weird. And it is STILL WEIRD. Specifically, the Podcasts show the (+) copy icon and the green border when I drag them over, but they don't show up in a list in the overlay. BUT -- I noticed that the purple audio bar on the bottom grew and I noticed that the sync icon showed up in the Finder window behind the overlap next to the IPOD with the icon (not the disk usage one)
So I then dragged a couple more over. I think I dragged about 6 in total. The purple audio bar kept growing. I clicked "Done" and then I ejected the IPOD and yes.... All 6 podcasts were playable on the Shuffle.
That's great, but how do I remove the podcasts in the future? How do I manage this?
Well -- there appears to still be a bug that doesn't show the contents in the overlay (at least for me)
I plan on clicking "restore ipod" to nuke the contents and then carefully dragging them over one at a time. (First episode ends up last in the playlist, FWIW.)
I hope this helps others who have this question, and I hope someone from Apple notices this and addresses the bug in the overlay.
I should add that this is an Orange 2nd Generation 1GB Shuffle.