It would be against my better judgement too - not that I'm often accused of having any!
It has been done, but I don't know how successfully. There are a number of potential problems, of which two immediately spring to mind:
If you compare the shape of the hull of your trawler with that of sailing boats, you'll notice a big difference, so at best you're likely to end up with a motor boat with sails, rather than even a true motor sailer.
Stability is another potential problem. Motor boats tend to rely on form stability rather than ballast, and only have enough to be stable in their designed role. Add a mast, rigging and sails, then a big squall that take you unawares, and things could get very hairy.
I think it would be a good idea to get a marine architect to look at this project before you spend any money on it