W/ APO, I can see wp-admin pages from fresh private mode

I purchased APO to speed up my websites, but then, I can now see wp-admin pages when I just come to my website from private mode (or in-cognito mode). I guess this is because of caching, and wonder if there is any way to disable that functionality.

I have a membership site, and now wp-admin, user specific cotents are shown to non-logged in users, while members complain that they cannot see their account page (which means their browser sees that they are not logged-in). Guess cookies are not stored, or mixed, maybe?

I turned off APO for now, and still would like to turn back, given the the amazing speed boost, but price that I have to pay, given what’s described above, feels way more than $5/month. Any potential fix?