I tried virtually every podcast software in the Android market.
I prefer Acast2 and it only costs a couple of dollars (it's been a while so I don't remember the exact amount).
There is a free Acast as well that you may want to try.
I used the free one a couple of times and then quickly switched to the paid version. (I hate ads). It is very impressive software and the player has large buttons that can easily be pressed.
Acast2 should meet all of your requirements. One additional need I had was to be able download a podcast for which I actually pay money. Acast2 could easily do it. I wasn't able to locate any other android software that could handle the same paid podcast.
The software is phenomenal and ranks with google sky map and Celeste as my favorite programs.
Google's Listen podcast program was quickly deleted off my phone.
Finally, the developer has a blog at ACast's Blog