Close, I think. Educated speakers of British English used to pronounce many words that started wh- as hw-. My father certainly did so, and you can hear it in old films from the thirties, forties and fifties. "Just hwat do you think you're doing, old boy?". Just purse your lips and make a soundless whistle, and you have it..
Yeah, there are some dialects of Swedish where it is pronounced as an "sh" sound. I've heard it pronounced that way on a Linköping radio station.
This pronunciation, though, sounds to me like more of an English "wh" sound, but pronounced further back in the mouth.
I first noticed this pronunciation while watching some Swedish television shows, such as "Bron" and "Wallander" (Swedish version)
Words like "själv" sounded more like "whelv"
dom is a colloquial spelling, but we accept that too.
However, please note that the "type what you hear" exercises have a bug where it'll only accept de and not dom. It's been reported to the developers but we have no idea when it'll be fixed. It's been like that forever.