The 6300 is fine to work with the RX 560 but if you already have the R9 270 then essentially this would be a sidegrade and a massive waste of $100. Save up the extra $30 and get the RX 570 4gb as that would be a good upgrade from the R9 270. The EVGA 600B should be ok with the RX 570 but if you do invest into a new PSU then get a good quality 550W as that would be better for upgrades. Also check you CPU temp, if it's thermal throttling (>60C) then getting a better GPU won't help increase performance much.Yes that’s my current built. But, I’m currently building the M5A78L-M PLUS/USB3 now. I plan to run the FX6300 stock cpu fan for this mobo. Bad idea for the RX560?
The 560 is indeed an entry level card, which means not much faster than your 270. You really need to jump up at least to a GTX 1050 Ti or equivalent. That means a GTX 960 or AMD 7950/7970 or AMD 280/380 types of cards. Of new cards the RX 570 is the sweet spot for gamers on a budget and looking to upgrade.
Be mindful of the power demands. The Corsair CX line is their cheap line, so having a bit more power than you need would be helpful with those PSUs. You don't want to run one of those at high load, it won't last.