Still no synch with my Exchange server.
There have been changes in how Mail stores it's data from previous versions of macOS. I'm not sure how you brought your data over to the new Mac, but the upgrade process could be at fault with your sync issues.
If you did not migrate your previous Mail data over, remove the Exchange account and add it back.
If you did use migration, let's look into the User's Library/Mail folder.**
Sierra stores your mail in V4. Do you see V2 of V3 folders? El Capitan used V3, previous versions macOS used V2. If yes, is there anything in these folders except MailData folder? If no, delete these folders.
Next, look in the V4 folder. You should see something similar to this:
(Note: each account will have it's own folder and the "On My Mac" folders will also have it's own folder. This example is for one account and the "On My Mac" folders)
C7AA1753-CDA7-4C5D-9505-05B73B565A50/
C9A984AA-A145-4988-8279-B93445951DAB/
MailData
Look in the alphanumeric folders to locate your Exchange account. With Mail quit, delete the Exchange alphanumeric folder.
In MailData folder, delete the three files with envelope in the name.
Log out under the Apple in the Menu bar.
Log in and empty the trash.
Open Mail. Mail will reindex your data.
When indexing is complete, Mail will open and should start the Exchange download again.
Let me know if this helps.
**Show User's Library
The User's Library folder is hidden by default. To unhide: Select the Finder in the Dock. Under Go in the Menu bar > hold down the Option key and you’ll see the Library.
Finding the User Library Folder
http://www.takecontrolbooks.com/resources/0167/site/chap11.html#FindingtheUserLi braryFolder