From what I can see, this means I need to exit the app to take some action with my mastodon account.

I haven’t tried posting anything yet, but just liking and boosting has been doing it through my Mastodon account. If I’m interacting with a bluesky post, then it uses the bluesky account that it automatically created for me. But since I signed in with my Mastodon account, all of those actions have been done by my Mastodon account.