Marco has been explaining the shortcomings in the ATP podcast in the last few weeks. Accidental Tech Podcast: 241: Jazzercise Boot Camphttps://overcast.fm/+CdS4p6Hg
He explains on his site that until Apple writes more into the API that an actual podcast app will not work reliably. https://marco.org/2017/09/24/what-watch-podcast-apps-need
His overcast app is by far my favorite of all time and he isn’t the only one trying but in my opinion if Marco can’t make it happen with the current code then no one can.
Apple's podcast app. If you haven't tried it in a long time it's actually very good now. I used to use Overcast, and I actually prefer Apple's now.
edit: Sorry, I realize now you meant an actually app you install on the watch. I don't think Apple actually has a watch podcast app. The integration works well though.
I suggest you clear the stored podcasts if it's using too much memory. Its in the settings under "Storage & Data Use".
Pocketcasts, Overcast are similar so it comes down to which you prefer the look and design of. Overcast also costs a lot more if you want to remove adverts though.
The watch apps from each don't allow storage on the watch. they will allow you to switch between podcast episodes playing on your iPhone. That's about it.