whatever you do, you're going to need to get a new motherboard which will cost some bucks. I would recommend staying intel. For high settings in witcher 3, you want better than a GTX 970 (that nvidia hairworks drops it to below 60 fps). If you want to break-the-bank bad graphics upgrade (and guarantee oldrim will run at max settings), you need a 1080 ti. The card WILL set you back almost $800 depending on where you buy it from and what manufacturer, over half your budget. With that card, I would echo what others say, get a platinum rated PSU (I have a Corsair HX860i (860 watts and modular) which was a RMA upgrade from a HX 850i (modular 850 watt PSU). Both platinum rated. Former is a replacement due to a outta spec 3.3 rail which caused ASUS anti-surge reboots. Cost for the express RMA was $230 to give an idea of PSU costs.). Fifth gen intel (socket 2011v3) should be able to be on win 7 if it's still being sold. Maybe 6th gen. 7th is a possibility. I could give suggestions but you'd be breaking the bank. Re-use your optical drives if they are recent enough, get a new case, get aircooling (aftermarket fans will work). Get a Noctura heatsink (probably as good as watercooling).
my PC specs (built in 2015)
CPU: intel i7-5930k
heatsink: Noctura NH-D15s
motherboard: Asus X-99 deluxe w/u3.1
RAM: 32 GB DDR4 RAM (forget who made it off top of my head)
storage: you don't want to know but it's a lot (2 TB SSHDD partitioned to win 7 and win 10 boot partitions if you need to know)
optical: Pioneer BD-XL drive and older Pioneer BD-burner
case: Rosewill rise (do NOT get this case as the third drive bay is mounted on top and makes it dangerous to put a drive there; I broke the SATA data cable on a drive moving a PSU around after mounting it wrong. The case has a bottom vent that is not ideal for a large PSU. That bad mounting was responsible in part for my first RMA. This is my second.) I would go with a NZXT case.
3x120 mm fans for front aftermarket. 1x 140mm fan top that needs replacing. 1x 80 mm fan covering the 2nd PSU slot.
monitor: Asus VG248QE 3d ready monitor
sound: onboard 7.1 but hooked up to a 7.1 headset bought this year.
GPU: EVGA GTX 1080 ti FTW 3 GAMING ($813 after taxes FYI)
With the current fan configuration, my CPU can get package temps of 47 celsius (each core sub 39). Ram hits about 39. GPU hits 36-37 with precision xoc off (much lower on).