So the RX 480 is a little disappointing due to overly high expectations but it is still the best card you can get in this price range, at least until prices shift in response to the new cards. I'm guessing the 480 is ballpark around as good as a GTX 970 depending on the exact game and DX version you are talking about. Good luck finding a 480 in stock at a reasonable price in the short term.
The GTX 950 is a bit light for "high-extreme settings @ FHD" and the 960 doesn't have much more grunt than the 950 so IMO neither meet your stated goal.
The GTX 970 would get the job done as would the RX 480. The 480 looks especially strong in DX12 titles.
I think it is worth waiting a bit longer to find a RX 480 in stock or maybe even a bit longer to see what nVidia's lower priced 14-16nm offerings look like. A GTX 1060 might be just the thing you need. July 7th is said to bring news of the 1060...