Cecilia,
I have found that the cogged belts need to be a wee bit tighter than the black V-belts to not slip. I think the plastic they are made from is slicker than the black belts. Also make sure the motor pulley and hand wheel belt grove is clean and free from oil or old belt residue.
I just converted my ALDENS Magic ZZ from a non-adjustable motor that required a rubber band belt, to an adjustable motor and put an amber cogged belt on it. It slipped like mad until I removed the belt and used some 600grit sand paper to clean the crud out of the motor pulley. Now it works OK.
I have never had a cogged belt stretch like what your LSMG said, a little yes, but not that much. I don't think there is any way you're going to get the new belt on without adjusting the motor for it. I wouldn't worry about moving the motor, that's why there is an adjustment to it.
Joe