I see this swap as dance where you change partners during the dance, like an old fashioned square dance. My thought is we start off with a set number of players, at least 12 for a full year or 6 for just half the year. Or we could go with the same partner for 2 months and then change. This could either start at the beginning or middle of next year. I will leave it up to the group to decide.
The idea is we have 12 (or 6) players. Everyone signs up in the database. When the swap closes, two lists will be made up with partners. Once the time is up, you would then rotate down the lists for a new partner, thus working your way down the list so that you have a chance to swap with everyone. I know this sounds a little strange but once it gets going it will work out.
In order for it to work, everyone would need to choose a theme or color to work with. The block would be up to the receiver. Or you could go scrappy with the color/theme.