600w is a good power rating, but much more important than that is the quality of the PSU. What brand and model do you have? (I say quality because there are many time-bomb type PSUs on the market that claim a power rating yet cannot deliver it. Worse yet they have so much noise and spikes that they damage the rest of your system... especially when they die.)
That said, the 1060 6GB is still a great 1080p offering for max graphics. as is the RX-580. The RX-480 is no slouch either although it's up and down with the 1060 depending on the graphics interface and game.
I would not recommend the 1060 3GB model. The differences don't stop at half the RAM. They include the same GPU cut down (like the RX470 is a cut-down RX480, but in the AMD case, with a designation that is clear that there is a difference.) Also, with 3GB of RAM, some games are hitting a wall in performance (and its only going to get worse as time goes on.)