"trolling" works. It's a method of fishing, just like trawling. It's one of the lesser-used definitions these days but IIRC that is the root for the more well known antagonizing definition.
Probably regional dialect. I spent my teenage years "trolling" for just about every part of the female anatomy, but never heard anyone use the word "trawling" in the same way. That said, trolling is a much more popular fishing method where I'm from.
In fishing isn't it a question of volume? IIUC trolling is moving very slowly/wakeless with a small number of lines, and trawling is a big net that catches everything?
Before 'trolling' became popular, 'trawling' was common and correct use for this case. Same kina use as fishing trawlers. The dictionary has more detail.
But trolling is also a type of fishing - you drag a line and try to catch people/fish on your bait (the phone call from the supposed son). That seems the perfect metaphor for what they were doing? They weren't dragging a big industrial net trying to catch anything at random passively (trawling) because they were trying to get individual people by calling them up one by one trying to get them to bite.
