It appears that I actually got the old integrated mail client working again!
The solution makes absolutely no sense, but I'll write it out in case someone happens to run into the same problem. I was just fiddling around with all the settings I could think of, and this is what worked for me. When you press the monkey wrench icon that appears next to your email address when you have selected one of your accounts, you can select "IMAP Folders". I unselected the inbox folder from there and then reselected it, after which everything was magically alright once again. With repeat and rinse for all the other accounts I have, naturally. Go figure, but I'm glad I can continue using my favourite mail client.
Thank you for all your help! Have a great summer!