Agree with Rolli59: The R9 280X is a power hungry card and I'd hate to see your system get damaged by stressing the existing PSU too much, so the GTX760 is your best option unless you add a better power supply to the upgrade.
If you DO decide to ugrade the PSU, a decent 600W unit would drive the current system (plus R9 280X) easily enough and I'd personally look for something a little more upmarket than the CX series. A Few Good Names: Corsair (not the CX!), Silverstone (not the Strider Essential), Antec, Seasonic, EVGA and XFX.
Have a look at some card reviews, Toms, Guru3D and Techpowerup are all good places to look because they compare plenty of cards in their benchmarks, that way you can see if the extra cost of a PSU+R9 280X is worth it over the simpler GTX760 upgrade.
Personally, I don't think swapping motherboards will help too much, the CPU is not likely to overclock enough to justify the cost and time required.