I think you've probably got the reason right for wearing them. Dawson was scrum half, and Thompson was hooker, so between both of them they were responsible for either feeding the ball in and out from the scrum or generally in open play, or, for throwing the ball in from a lineout.
Can't say I've noticed anybody else wearing them, but then again I haven't really been looking. I think Clive Woodward instilled absolute professionalism in that team, so maybe it was a psychological thing in terms of personal game preparation ? Who knows ?
Hope you're keeping well mate.