Is there a way to "he put on his coat" without actually using the verb "put on"? A friend suggested "he dressed his coat" but it sounds very strange to me.
to put on (an article of clothing)
You might consider something more descriptive:
He threw on his coat.
He slipped on/into his coat.
He pulled on his coat.
He flung on his coat.
If you're going for the most neutral verb possible, 'don' is, as other answers/comments have indicated, a correct (and probably comedically archaic) usage.