With the given budget, i would build a new computer.
Keep the GPU and the HDD.
PCPartPicker Part List
CPU: AMD Ryzen 7 5800X 3.8 GHz 8-Core Processor ($439.00 @ Newegg)
CPU Cooler: Corsair iCUE H150i ELITE CAPELLIX 75 CFM Liquid CPU Cooler ($169.99 @ Newegg)
Motherboard: MSI MAG B550 TOMAHAWK ATX AM4 Motherboard ($168.83 @ Amazon)
Memory: Corsair Vengeance RGB Pro 32 GB (2 x 16 GB) DDR4-3600 CL18 Memory ($229.99 @ Amazon)
Storage: Samsung 970 Evo 500 GB M.2-2280 NVME Solid State Drive ($79.99 @ B&H)
Case: Corsair 4000D Airflow ATX Mid Tower Case ($94.99 @ Amazon)
Power Supply: Corsair RMx (2018) 850 W 80+ Gold Certified Fully Modular ATX Power Supply ($217.52 @ Amazon)
Case Fan: Corsair ML120 PRO 47.3 CFM 120 mm Fans 3-Pack ($129.99 @ Amazon)
Total: $1530.30
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2021-04-17 10:46 EDT-0400
5800X is a 8 core 16 thread cpu, that is considered to be one of the best for gaming.
Will not struggle with any gaming load you put on it.
8 core 16 threads should last you well over 5 years also.
I put a 360mm AIO in here since the 5800X is known for getting a bit toasty, plus the AIO looks awesome.
The motherboard has solid VRM's, tons of features and out of the box compatability with 5000 series cpu's.
2x16gb should be plenty for a normal user. Its a bit overkill for gaming, but if you dont plan on upgrading any time sooner, its perfect.
Added a fast m.2 in there for Windows and related programs.
Changed the case since your existing one only has a front and rear fan mount.
This one looks better imo, has better airflow and its overall newer and more modern.
Changed the PSU since your existing one is really bottom tier and would honestly only last with your GTX 960.
Speaking of GTX 960, i would keep that until you can get a newer GPU for a normal price.
Something like a RTX 3060Ti would be pretty nice with this build.
This upgrade would sky-rocket both your FPS and overall user experience.
You can sync up the rgb of the fans of the radiator, the 3 fans that i added and ram in Corsair's iQue software and make it look even cooler.