This is probably two months after your event, but you can have 5 rounds of foursomes and still have all partners different.
[TABLE="width: 760"]
<colgroup><col><col span="24"></colgroup><tbody>[TR]
[TD][/TD]
[TD="colspan: 4"]Foursome 1[/TD]
[TD][/TD]
[TD="colspan: 4"]Foursome 2[/TD]
[TD][/TD]
[TD="colspan: 4"]Foursome 3[/TD]
[TD][/TD]
[TD="colspan: 4"]Foursome 4[/TD]
[TD][/TD]
[TD="colspan: 4"]Foursome 5[/TD]
[/TR]
[TR]
[TD]R1[/TD]
[TD]A1[/TD]
[TD]A6[/TD]
[TD]B1[/TD]
[TD]B6[/TD]
[TD][/TD]
[TD]A5[/TD]
[TD]A10[/TD]
[TD]B5[/TD]
[TD]B10[/TD]
[TD][/TD]
[TD]A4[/TD]
[TD]A9[/TD]
[TD]B4[/TD]
[TD]B9[/TD]
[TD][/TD]
[TD]A3[/TD]
[TD]A8[/TD]
[TD]B3[/TD]
[TD]B8[/TD]
[TD][/TD]
[TD]A2[/TD]
[TD]A7[/TD]
[TD]B2[/TD]
[TD]B7[/TD]
[/TR]
[TR]
[TD]R2[/TD]
[TD]A5[/TD]
[TD]A9[/TD]
[TD]B3[/TD]
[TD]B7[/TD]
[TD][/TD]
[TD]A4[/TD]
[TD]A8[/TD]
[TD]B2[/TD]
[TD]B6[/TD]
[TD][/TD]
[TD]A3[/TD]
[TD]A7[/TD]
[TD]B1[/TD]
[TD]B10[/TD]
[TD][/TD]
[TD]A2[/TD]
[TD]A6[/TD]
[TD]B5[/TD]
[TD]B9[/TD]
[TD][/TD]
[TD]A1[/TD]
[TD]A10[/TD]
[TD]B4[/TD]
[TD]B8[/TD]
[/TR]
[TR]
[TD]R3[/TD]
[TD]A4[/TD]
[TD]A7[/TD]
[TD]B5[/TD]
[TD]B8[/TD]
[TD][/TD]
[TD]A3[/TD]
[TD]A6[/TD]
[TD]B4[/TD]
[TD]B7[/TD]
[TD][/TD]
[TD]A2[/TD]
[TD]A10[/TD]
[TD]B3[/TD]
[TD]B6[/TD]
[TD][/TD]
[TD]A1[/TD]
[TD]A9[/TD]
[TD]B2[/TD]
[TD]B10[/TD]
[TD][/TD]
[TD]A5[/TD]
[TD]A8[/TD]
[TD]B1[/TD]
[TD]B9[/TD]
[/TR]
[TR]
[TD]R4[/TD]
[TD]A3[/TD]
[TD]A10[/TD]
[TD]B2[/TD]
[TD]B9[/TD]
[TD][/TD]
[TD]A2[/TD]
[TD]A9[/TD]
[TD]B1[/TD]
[TD]B8[/TD]
[TD][/TD]
[TD]A1[/TD]
[TD]A8[/TD]
[TD]B5[/TD]
[TD]B7[/TD]
[TD][/TD]
[TD]A5[/TD]
[TD]A7[/TD]
[TD]B4[/TD]
[TD]B6[/TD]
[TD][/TD]
[TD]A4[/TD]
[TD]A6[/TD]
[TD]B3[/TD]
[TD]B10[/TD]
[/TR]
[TR]
[TD]R5[/TD]
[TD]A2[/TD]
[TD]A8[/TD]
[TD]B4[/TD]
[TD]B10[/TD]
[TD][/TD]
[TD]A1[/TD]
[TD]A7[/TD]
[TD]B3[/TD]
[TD]B9[/TD]
[TD][/TD]
[TD]A5[/TD]
[TD]A6[/TD]
[TD]B2[/TD]
[TD]B8[/TD]
[TD][/TD]
[TD]A4[/TD]
[TD]A10[/TD]
[TD]B1[/TD]
[TD]B7[/TD]
[TD][/TD]
[TD]A3[/TD]
[TD]A9[/TD]
[TD]B5[/TD]
[TD]B6[/TD]
[/TR]
</tbody>[/TABLE]
This would also allow you to have 4 ability groups, since each foursome has 1 player each from the groups A1-A5, A6-A10, B1-B5 & B6-B10.
GM