Wirelessly posted (BlackBerry: BlackBerry8530/5.0.0.1030 Profile/MIDP-2.1 Configuration/CLDC-1.1 VendorID/417)

"I read posts about people going to Asia and South America and places like that using these free tickets (so the flights must be BA only)."

That's where you're going wrong. As already noted, award tickets using Avios do not have to be on BA flights. But if you want to use a two-for-one voucher, then you are restricted to BA flights.

For a list of BA partner flights that have little or no fuel surcharges, search for the thread title in this forum with the phrase, "no fuel surcharge".