Your 1200W power supply is sufficient enough. if of course it's a proper brand with the right power ratings.
Your PSU should have the sufficient connectors on it , I guess if it's not too old.
If it doesn't then you need to get a power adapter from any of the online shops or your dealer that will comply with the power requirement of the 5970 and provided that you have the other power cable free from your PSU for it to comply with.
You do not need the power adapters. Just use one 8 pin (6+2) and one 6 pin each.
I do not know where the original specs for the PCIe power cables came from, but look at the 6 and 8 pin connectors. The 6 pin has 3 hot (yellow) wires and 3 ground (black) wires. The 8 pin has 3 hot and 5 ground wires. So each will carry the same amount of current through the 3 hot wires.