In my experience, a 12 Team Dual tournament can be run one of two ways:
Option A: Set up 3 pools of 4 teams each: (These teams were chosen purely out of familiarity with their districts and nothing should be read into)
EXAMPLE: Pool A- Pike, North Central, Lawrence North, Lawrence Central
Pool B- Ben Davis, Decatur Central, Perry Meridian, Southport
Pool C- Warren Central, Franklin Central, New Palestine, Greenfield Central
You then run 3 rounds, dueling all 3 other teams in your pool: (Round 1: Pike vs NC, LN vs LC. Rd 2: Pike vs LN, LC vs NC, Rd 3: Pike vs LC, NC vs LN)
Keep track of records, use standard pool tiebreakers (most points scored, etc;) and place the teams 1-4 in each pool.
Then, place each first place team into a pool of 3 and each second place team, etc;
So If Pike, BD, and WC all won their pools, they would be in the same pool together (the championship pool): Then you have each team dual the other two teams.
NOTE: In this plan, you get a true champion, but you have to have it last an extra round because of byes. You'd run 6 mats during rounds 1-3, but during rounds 4,5, and 6, you'd have 4 teams on bye, which seems like a waste of time, but it's the only way to ensure everybody wrestles 5 times and a true team champion is determined.
Option B: Set up exactly what Trackgopher has done and instead of 3 pools of 4, do 4 pools of 3 (still have the bye problem) and then have 3 pools of 4 for placement. I believe this is what the IHSWCA Dual Championship tourney utilizes. Only 5 rounds. Still have four teams on byes for 2 of those rounds though and everybody gets only 4 duals instead of 5.
You can obviously do two separate pools of 6 as you mentioned as well. In any of these options, there will be strong wrestlers on weaker teams that don't get to face the better wrestlers on better teams, there will just be less of that in the first two options.
Good luck!