kos,
AR2209 is reputed to give a bit less recoil than AR2213 for equal velocities.
Usually you shoot the right barrel first but as another member here has pointed out, it often won't make much or any difference.
Shoot your second barrel with as little time as possible after the first.
This has nothing to do with regulation but is to allow your memory to duplicate the exact same sighting picture as with your first shot!
Sitting is often more steady than standing unless you have a proper double rifle standing rest.
Use sandbags or similar at the forend and just rest your elbows on the bench with nothing else touching the buttstock.
Hold the forend in your hand and rest your hand on top of the sandbag so the rifle is not in contact with anything other than your hands.

Questions.
What group sizes are you getting from each individual barrel?
What is the overall group size of both barrels and at what distance?
What load are you using and have you chronographed any of your loads?