Apple Mail: messages piling up in Drafts - a solution

The Drafts folder on one MBP had been giving us trouble lately:

multiple copies of messages kept piling up and never were deleted. This
was rather irritating because sometimes you weren't sure whether the
message had actually been sent.

After a while of googling I came across a post on the Apple user forum,
that had a transcript with Apple support.

They suggested to go into Messages > Take All Accounts Offline and
quit Mail.

Then open Mail again and select Messages > Take All Accounts Online

This seems to have fixed the problem for good.