For several reasons it it best to look for schedules with a multiple of 7 rounds. The primary motivation is to allow all players to have the same number of games, however it also makes the schedule easier to construct. So below I give a schedule for 14 rounds of play. Here the 14 players are numbered 0 to 13.
Table 1 Table 2 Table 3 Byes
(5 10 : 1 12) ( 3 13 : 0 9) ( 6 4 : 7 11) ( 2 8)
(6 11 : 2 13) ( 4 7 : 1 10) ( 0 5 : 8 12) ( 3 9)
(0 12 : 3 7) ( 5 8 : 2 11) ( 1 6 : 9 13) ( 4 10)
(1 13 : 4 8) ( 6 9 : 3 12) ( 2 0 : 10 7) ( 5 11)
(2 7 : 5 9) ( 0 10 : 4 13) ( 3 1 : 11 8) ( 6 12)
(3 8 : 6 10) ( 1 11 : 5 7) ( 4 2 : 12 9) ( 0 13)
(4 9 : 0 11) ( 2 12 : 6 8) ( 5 3 : 13 10) ( 1 7)
(2 5 : 3 4) (13 12 : 7 1) ( 9 11 : 8 0) (10 6)
(3 6 : 4 5) ( 7 13 : 8 2) (10 12 : 9 1) (11 0)
(4 0 : 5 6) ( 8 7 : 9 3) (11 13 : 10 2) (12 1)
(5 1 : 6 0) ( 9 8 : 10 4) (12 7 : 11 3) (13 2)
(6 2 : 0 1) (10 9 : 11 5) (13 8 : 12 4) ( 7 3)
(0 3 : 1 2) (11 10 : 12 6) ( 7 9 : 13 5) ( 8 4)
(1 4 : 2 3) (12 11 : 13 0) ( 8 10 : 7 6) ( 9 5)
I am sorry that I can't offer a solution to the 11 round problem, however I think you will find it easier to remove 3 rounds from above, rather than trying to add 4 new rounds to the 7 round schedule that you mentioned. If you can play the full 14 rounds then all players will have 12 different partners and never oppose the same player more than twice.