Problem moving messages using Office 365 / Exchange server and Mail.app

We recently switched our company from Google Workplace (Gmail) to Office 365 hosted email (Exchange). The switch was easy, migrate the mail, add the account to my mac, turn off the old account, turn on the new one, and voila, all the messages in my inbox now came from the O365 server and not the Gmail server.

But I've had a couple of issues crop up recently. First, I have rules that move various messages to various "on my Mac" folder and I suspect these warnings are because of those.

Second, and much worse, when I move emails from my combined inbox to a folder on my mac, often times the message ends up in the folder with zero content in it, just a blank email with the sender and subject showing, but nothing else. The body of the email shows all white. when I look at the raw message content, that is also blank.

Another thing I noticed is that when I move messages out of the inbox to a local Mac folder, and go back to the Office 365 online email client, they are still in the inbox. With Gmail, they would be deleted, not hang around to eat up storage space. Seems like as an IMAP server (isn't it?) it should be deleting the message when it is moved out of the IMAP inbox.

I've looked over the "exchange" server setting, not much there, certainly nothing that might cause this.

Anyone experienced either of these issues?