One suggestion that I have is that some teams will switch groups as part as off season process.
Maybe make it as a rule that 5th place teams will change groups.
One more chapter in the old 5th place is the best/worst story? :-) By fixing some position for the switch we would open the door for some more unsportmanlike tactics. What I would like to see is that every team generally tries to finish as high in the standings as possible.
One simple solution to slightly refresh the conferences would be to introduce a minimum of 2 new teams on both sides each season. This can be done by switching one of the relagation playoff winners to the other conference in case the 6th and 7th place teams of the same conference win their series. The problem with this is that the refreshing happens with potentially weaker teams (especially if the switcher is the 7th, although they did just beat the 6th of the other conference). In this case, the aim of balancing the competition would not really be achieved. Especially so for div I leagues.
Only personal opinion/disclaimer below (no need to discuss):
Random leagues don't create as much rivalry as the current system, which is why I am personally pretty much against
any reshuffling. We already have a random competition, which goes to the other extreme: the national tournament. And some users would like to make that much more predictable. I am not one of those, I really enjoy the tournament the way it is.