I can describe what I did previously, I think that could help you.
On my old Pixel phone I used Total commander and select from main page Installed apps. It listed all my installed apps, resp. apk files. I copied the files to download folder on the phone and than transferred them to my USB stick. (For some reason I couldn't copy the files directly, but that's not the issue.)
Then I transferred those apk files to new phone without Google play and Google play services. I installed them. That's it. Sometimes an app showed a notification that the app needed Google play services, but you can set it that the notification never shows again.
I know for sure that (at least older version of) Moon+reader worked flawlessly without any Google stuff, because I used it on my "degoogled" Lineage OS. I've never used Podcast addict so I can't comment on that.
I hope this helps.
Edit: Keep in mind that using this method, quite logically you wouldn't be able to update these apps via Aurora as the apps wouldn't be able to authorize paid content. If you want to update the paid app, you can either repeat the method and back up the new file, or use Google Play with your account, or contact the app developer.