I've had similar texture issues with FarCry 4 actually on my Fury. I honestly think it's the Crimson drivers as I didn't notice any issues beforehand. Crimson has done a lot of great stuff and I greatly prefer it to the old CCC, but it has had some flukes. Try reinstalling Crimson after stripping it out with guru3d's graphics driver removal tool (or AMD's), and if that doesn't solve it try switching to/from the Mantle API. You may also try different AA settings, but if all else fails, try rolling back the driver set.
Mantle? And it's the OLD 2004 Far Cry
Whoops, my bad, thought you were referring to FarCry4.
A lot fewer graphics options at least makes it easier to pin down.
Use a graphics driver removal tool like Display Driver Uninstaller to ensure all the old GeForce drivers are gone and reinstall the ATI ones.
Reinstall DX9 by running the DX9 installer from the Farcry install folder.
Turn off AA entirely and see if the problem appears, if not, keep bumping the AA till it's either maxed or the problem comes back.
Good luck, beyond this there are a few other things I could suggest, but these are likely the biggest things to try.