I've checked the webmail for all 4 accounts and they display messages as being unread, though Aquamail shows them as read.
I have the same accounts on a tablet in Aquamail and the messages display as unread, even after a refresh. Overnight, when this happens the most, the only device accessing mail at the same time is the tablet. Maybe I will delete the accounts in the tablet and set them up again to match the settings of the phone in case there is a difference in how the tablet is receiving the messages and reporting back to the server.
If I have 10 new messages in the inbox, not all 10 will be marked as read, the ones marked read are not grouped together even - they might be messages 2,3,6,7,8,10.
Since the problem didn't start until a fresh install of Aquamail on a reset phone, it has been confusing to track down - especially since it only affects this one device.
Thank you for your help.