I will say this: I do not want to see anyone dropping out and not doing the swaps just because someone didn't get to do one last time or doesn't get in. There has to be a way to do this to at least make most of those that want to do it happy at least some of the time.

Only way I can think of to make sure everyone gets in each at least once in a while is this:
Everyone signs up that wants to tie for next set of swaps
Swap meisters take the number of tiers they want
Everyone ties their jigs
Those who don't get in go to top of list for next time.
When everyone has tied, a new list is made, either by order of sign up or by random.

I do see the drawbacks:
Could lead to a LOT of swaps going on at one time.
Not enough meisters, people still have to wait but at least get to get into one next time

I do know this: It would be a shame to start losing members because of not getting into a swap.