Unfortunately there will be no balanced schedule for your 6 man teams.
If you had 16 players, then you could consider
something like this.
If each of the 12 players had 6 games each, then it is possible to arrange it so that all players oppose each other exactly two times. Of course 6 games each does mean partnering with one of you own team members twice.
(A2 A3):(B6 B1)
(A4 A1):(B6 B4)
(A5 A6):(B4 B1)
(A6 A4):(B6 B5)
(A6 A2):(B3 B1)
(A5 A2):(B6 B3)
(A1 A6):(B6 B2)
(A2 A4):(B5 B4)
(A1 A4):(B1 B3)
(A1 A2):(B2 B4)
(A6 A3):(B4 B3)
(A5 A4):(B3 B2)
(A3 A1):(B5 B3)
(A3 A5):(B4 B2)
(A3 A5):(B6 B5)
(A5 A1):(B1 B5)
(A2 A6):(B2 B5)
(A4 A3):(B1 B2)
and note that it is not possible to arrange these 18 games in to 6 rounds of 3 foursomes, although 9 rounds of 2 foursomes should be possible.