Should I upgrade my current PC or build a new one?

First, getting slower might simply be a software issue, try to see if you have a lot of unwanted program that you can delete. Second, clear out your browser caches and such. Last, open it up and use a can of compressed air to clean out all the dust and dirt. Sometimes it can heat things up and cause the computer to slow down. You usually can find it at a local computer store.

http://www.newegg.com/Product/Product.aspx?Item=1DP-0003-00001&cm_re=compressed_air-_-1DP-0003-00001-_-Product

If you want to spend money you might want to junk the pre-built. Rather than trying to salvage anything out of it I would just try to sell it as a unit or use it for something else.

The i5 build is the way to go, much faster and better than the FX 6300. I loaded your proposed build in PCPartPicker and modified it a bit. I changed the CPU to a 4460, slightly slower not likely to affect the performance of the build. Added an SSD to load OS and programs on, it make a big difference in boot times and screen changes while gaming. Changed your 960 to a 380, bit better card and you are a little low on the GPU end of things. Rule of thumb is 1.5-2 times the amount you spend on the CPU. Last, changed the PSU to a better one, it does matter. You don't need a 650W but it is an okay match, gives you some room for expansion and most importantly, it was the cheapest good one that I saw. Hope this helps.

Current Build
PCPartPicker part list: http://pcpartpicker.com/p/k6vxXL
Price breakdown by merchant: http://pcpartpicker.com/p/k6vxXL/by_merchant/

CPU: Intel Core i5-4590 3.3GHz Quad-Core Processor ($188.75 @ OutletPC)
Motherboard: ASRock H97 Anniversary ATX LGA1150 Motherboard ($74.99 @ SuperBiiz)
Memory: G.Skill Ripjaws Series 8GB (2 x 4GB) DDR3-1600 Memory ($29.99 @ Newegg)
Storage: Western Digital Caviar Blue 1TB 3.5" 7200RPM Internal Hard Drive ($49.89 @ OutletPC)
Video Card: Asus GeForce GTX 960 4GB Video Card ($209.19 @ OutletPC)
Case: Deepcool TESSERACT SW ATX Mid Tower Case ($47.98 @ Newegg)
Power Supply: Corsair CX 600W 80+ Bronze Certified Semi-Modular ATX Power Supply ($44.99 @ Newegg)
Optical Drive: Asus DRW-24B1ST/BLK/B/AS DVD/CD Writer ($15.99 @ Newegg)
Wireless Network Adapter: Gigabyte GC-WB867D-I 802.11a/b/g/n/ac PCI-Express x1 Wi-Fi Adapter ($29.89 @ OutletPC)
Total: $691.66
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2016-02-04 16:19 EST-0500

Proposed Build
PCPartPicker part list: http://pcpartpicker.com/p/Lgs4yc
Price breakdown by merchant: http://pcpartpicker.com/p/Lgs4yc/by_merchant/

CPU: Intel Core i5-4460 3.2GHz Quad-Core Processor ($174.89 @ OutletPC)
Motherboard: ASRock H97 Anniversary ATX LGA1150 Motherboard ($74.99 @ SuperBiiz)
Memory: G.Skill Ripjaws Series 8GB (2 x 4GB) DDR3-1600 Memory ($29.99 @ Newegg)
Storage: A-Data Premier SP550 240GB 2.5" Solid State Drive ($59.99 @ Amazon)
Storage: Western Digital Caviar Blue 1TB 3.5" 7200RPM Internal Hard Drive ($49.89 @ OutletPC)
Video Card: Sapphire Radeon R9 380 4GB NITRO Dual-X OC Video Card ($208.98 @ Newegg)
Case: Deepcool TESSERACT SW ATX Mid Tower Case ($47.98 @ Newegg)
Power Supply: Antec EarthWatts Green 650W 80+ Bronze Certified ATX Power Supply ($54.99 @ Newegg)
Optical Drive: Asus DRW-24B1ST/BLK/B/AS DVD/CD Writer ($15.99 @ Newegg)
Wireless Network Adapter: Gigabyte GC-WB867D-I 802.11a/b/g/n/ac PCI-Express x1 Wi-Fi Adapter ($29.89 @ OutletPC)
Total: $747.58
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2016-02-04 16:52 EST-0500