And yeah, Jessie is the most dramatic of the bunch, so she may or may not stand by what she said. I think these might be some of the likely scenarios:
1. Jessie was serious and stands by what she said, so she goes to Isshu along with the rest of TR. There are contests, Dawn joins up with Ash at episode 4 or 5 because it turns out that "Contest Season" in Kanto is almost done, and there are very few ribbons left to take. She joins up with Ash because of Isshu's Contest Season just starting. Jessie/James get the job of food salesmen(like in Hoenn league) in every contest.
2. Jessie was serious and stands by what she said, the writers thought accomodating Jessie without contests would be hard, so they don't include them and thus, no Dawn. Perhaps Pokeathlon takes it's place? I mean, imagine that the new girl"s goal is to become Pokeathlon Champion?
3. Jessie doesn't stand by her choice and joins contests in Isshu. Then, Dawn would have no reason to leave the group.