Zac,

Pasted below is a schedule for 15 rounds. Each team except one plays once per round. Each team plays every other once. The exceptional team has a bye. Use the schedule twice. For randomness, if you want it, shuffle the rounds the second time.

In what follows there are 15 teams, corresponding to the 15 positions in each sequence. The way to read the schedule is that the positions in each sequence are the teams, and the numbers indicate the pairings. Zeros indicate the bye team. On Day 1, team 1 has a bye, then 2 plays 3, 4 plays 5, and so on. On day 2, team 2 has a bye, 1 plays 3, 4 plays 6, etc.

Just use three rounds per week to get the schedule you are looking for. Most teams play 3 games per week. In each week there would be 3 teams that play 2 games.

Day 0 : (0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7)
Day 1 : (1, 0, 1, 2, 3, 2, 3, 4, 5, 4, 5, 6, 7, 6, 7)
Day 2 : (1, 1, 0, 2, 3, 3, 2, 4, 5, 5, 4, 6, 7, 7, 6)
Day 3 : (1, 2, 3, 0, 1, 2, 3, 4, 5, 6, 7, 4, 5, 6, 7)
Day 4 : (1, 2, 3, 1, 0, 3, 2, 4, 5, 6, 7, 5, 4, 7, 6)
Day 5 : (1, 2, 3, 2, 3, 0, 1, 4, 5, 6, 7, 6, 7, 4, 5)
Day 6 : (1, 2, 3, 3, 2, 1, 0, 4, 5, 6, 7, 7, 6, 5, 4)
Day 7 : (1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7)
Day 8 : (1, 2, 3, 4, 5, 6, 7, 1, 0, 3, 2, 5, 4, 7, 6)
Day 9 : (1, 2, 3, 4, 5, 6, 7, 2, 3, 0, 1, 6, 7, 4, 5)
Day 10 : (1, 2, 3, 4, 5, 6, 7, 3, 2, 1, 0, 7, 6, 5, 4)
Day 11 : (1, 2, 3, 4, 5, 6, 7, 4, 5, 6, 7, 0, 1, 2, 3)
Day 12 : (1, 2, 3, 4, 5, 6, 7, 5, 4, 7, 6, 1, 0, 3, 2)
Day 13 : (1, 2, 3, 4, 5, 6, 7, 6, 7, 4, 5, 2, 3, 0, 1)
Day 14 : (1, 2, 3, 4, 5, 6, 7, 7, 6, 5, 4, 3, 2, 1, 0)

Hope this helps.
—Victoria