Correct -- you wouldn't need to use a Red Sox season of one player and a Yankees season of the other.
This brings up a good question though that likely needs clarification. Since we are only using 1920 - present, what happens in cases like Babe Ruth, where he played for a franchise before 1920 but not after 1920?
In these cases, the player would not be eligible for the earlier franchise. So, for Ruth, he would only be eligible for Yankees / Braves.