I have to agree with David, it's a completely different regional usage than in the western US as well. Here in Seattle, home to a very large fishing fleet, trawling is used exclusively to refer to towed nets, often with otter boards, and usually by vessels between 20 and 100 meters in length that go out to catch cod, haddock, shrimp and rockfish. Dragging is used on occasion to refer to the same method, but not to the extent as in such New England ports as New Bedford or Gloucester.
Maritime history, especially that of our local fishing fleet, is a personal pastime of mine, and in all my research I haven't come across any local usage of the word here in the way you describe. Instead, what you mention is closest to long-lining here, where single, detachable lines with a series of baited hooks are left in place, either buoyed or anchored, to be retrieved later. These vessels often operate here in the halibut and cod fisheries.
Kind Regards,
Kyle