I'll give you some of my experience... I was running a GTX 970 and a I5 3570K and playing Battlefield 4 at 100 FPS on medium settings and when Battlefield 1 came out I had a very hard time running it over steady over 60 FPS on a 1080p monitor. The GPU was loafing and but the CPU had all 4 cores 100% utilized with wild swings of frame rates between 50 and 90 FPS. G Sync didn't help, a mild over clock helped and took away the worst of it but the lag was getting me killed at odd times. The motherboard finally died and I upgraded to the I7 6700K. I now could +100 FPS on medium settings and the game played well.
I recently picked up a GTX 1080 on sale for $429 and now Battlefield 1 runs at 144hz on ultra settings. The Beta for Battlefield V runs the same. On the plus side, and the reason I went to a GTX 1080, is that I can now play Elite Dangerous in VR at high settings.
So, while a new graphics card will definitely help you out, in the long run, you will need to budget for a better CPU. Personally, based on my experience, I would be hard pressed to decide which to do first. The GTX 970 is still very capable, especially if you're only running a 60hz monitor. Depending on the games you play and the monitor refresh rate, the CPU might be a better upgrade. Another thing to consider is the getting a GTX 1070 or Vega 56, these will be excellent cards if you're not running a 144hz monitor but even then they will likely be all that you need. That would leave a nice bit of money to put towards upgrading that CPU. I hope this helps.
Windows10, EVGA G2 750w Power Supply, Acer XB270H G-synch monitor, MSI Krait Gaming 3X, I7 6700K, 16 gigs of CORSAIR Vengeance LPX DDR4 3200 MHz ram, Crucial 500 gig SSD, EVGA GTX1080 FTW