Splitting up the party on separate flights is one thing, and may succeed in getting past the (likely, but not certain) problem of ever finding seven saver award seats on the same flights for a single booking.

Separate tickets on separate carriers is a much higher level of risk. Schedule changes, late operations, baggage handling (since Southwest doesn't interline), in my opinion, make that a very, very bad plan for a group of seven.

I might suggest that when you can't get seven award tickets, try for 5 (or 4, or 3) and buy the rest to get all on the same plane. There are airport transportation advantages and rebooking strengths from that (a little agent sympathy can go far). I trust that you all enjoy each other's company since you're traveling together!