Where are podcast playback settings?
Where are the settings on iphone iOS10 to change how podcasts play?
Specifically, I'd like to not have a new podcast start as soon as one ends.
Do you still have the crescent moon icon at the bottom when viewing the currently playing episode? That has the option to 'stop when episode ends'. It may allow you some control via the sleep timer.
Investigate the settings in Settings > Podcasts to see if there is anything to help there. NOTE: I don't have iOS 10 yet.