The following schedule may be useful if you have 12 players. Although you asked for 22 weeks I think 21 weeks is a better choice since then it is possible to have people play 7 times on the hard court, 7 times on the har-tru court and have 7 byes. Perhaps you could rank the women after 21 rounds and have a play-off for prizes in the 22nd week. In terms of the doubles play, all pairs of players partner each other at least once and at most twice, and all pairs of players oppose each other at least twice and at most three times. The schedule has come from a computer program which unfortunately has no way of spreading the byes evenly through the 21 weeks, but you could rearrange the order of the 21 rows to improve this.
hard har-tru byes
(2 7 v 4 9) ( 3 6 v 11 8) ( 5 10 12 1)
(3 8 v 5 7) ( 1 4 v 12 9) ( 6 11 10 2)
(1 9 v 6 8) ( 2 5 v 10 7) ( 4 12 11 3)
(8 7 v 4 10) (12 1 v 3 5) ( 6 2 9 11)
(9 8 v 5 11) (10 2 v 1 6) ( 4 3 7 12)
(7 9 v 6 12) (11 3 v 2 4) ( 5 1 8 10)
(6 2 v 12 5) ( 7 4 v 11 1) ( 8 10 3 9)
(4 3 v 10 6) ( 8 5 v 12 2) ( 9 11 1 7)
(5 1 v 11 4) ( 9 6 v 10 3) ( 7 12 2 8)
(7 11 v 6 4) (10 8 v 2 1) (12 3 9 5)
(8 12 v 4 5) (11 9 v 3 2) (10 1 7 6)
(9 10 v 5 6) (12 7 v 1 3) (11 2 8 4)
(1 2 v 11 5) ( 8 7 v 10 12) ( 9 3 6 4)
(2 3 v 12 6) ( 9 8 v 11 10) ( 7 1 4 5)
(3 1 v 10 4) ( 7 9 v 12 11) ( 8 2 5 6)
(1 7 v 2 9) ( 4 12 v 6 8) (10 5 11 3)
(2 8 v 3 7) ( 5 10 v 4 9) (11 6 12 1)
(3 9 v 1 8) ( 6 11 v 5 7) (12 4 10 2)
(3 12 v 10 11) ( 8 4 v 2 6) ( 1 9 5 7)
(1 10 v 11 12) ( 9 5 v 3 4) ( 2 7 6 8)
(2 11 v 12 10) ( 7 6 v 1 5) ( 3 8 4 9)
Hope that helps,
Ian.