The FX-6300 is definitely not worth keeping, IMO. Replacing that, however, means you need a new CPU, motherboard, and RAM (I think the AM3+ socket that you're using is a DDR3 system, and modern CPUs need DDR4).
If gaming is your main use and you aren't doing a significant amount of content creation (video editing, music-making, rendering, etc), I'd say to aim for Intel's new offerings. The thing is, Intel is expected to release new CPUs in the next month or so, and Nvidia is releasing new GPUs by the end of the month. You might want to wait for reviews for both of these releases before buying anything, seeing as how you seem to keep your PC for 6+ years at a time.
As a rough idea, look at builds within your budget range from the OP. Then, swap out whatever GPU is in that build for the a $700 RTX 2080 (one of the new cards coming from Nvidia) and that'll give you a sense of what you can buy using a newly-released GPU. You should be able to get a really incredible PC for that $1,500-$1,700 budget. Something roughly like the build below (which lacks a GPU), and then you can throw in the RTX 2080 for $700. This also includes Windows in the budget, so if you can transfer your current license, you can knock another $100 off. This is using one of the best CPUs currently available from Intel, but if they release a new series in the next month or two, it might be worth swapping out the CPU for one of those, depending on pricing, benchmarks, and how soon you want to build.
As a side note, these are somewhat placeholder parts; you can probably find a better case, PSU, etc, if you look around. You could also swap the CPU cooler for the NH-D15S (single fan instead of the dual-fan I've included) to save a bit more, and that'll be more than enough if you're not overclocking.
PCPartPicker part list /
Price breakdown by merchant
CPU: Intel - Core i7-8700 3.2GHz 6-Core Processor ($307.98 @ Amazon)
CPU Cooler: Noctua - NH-D15 82.5 CFM CPU Cooler ($85.22 @ Newegg)
Motherboard: *
MSI - B360-A PRO ATX LGA1151 Motherboard ($69.99 @ B&H)
Memory: *
G.Skill - Aegis 16GB (2 x 8GB) DDR4-3000 Memory ($135.98 @ Newegg)
Storage: *
SanDisk - SSD PLUS 240GB 2.5" Solid State Drive ($49.99 @ Adorama)
Storage: *
Western Digital - RE4 2TB 3.5" 7200RPM Internal Hard Drive ($54.95 @ Newegg Marketplace)
Case: *
Fractal Design - Focus G (White) ATX Mid Tower Case ($51.15 @ Walmart)
Power Supply: *
Corsair - CXM 550W 80+ Bronze Certified Semi-Modular ATX Power Supply ($42.98 @ Newegg)
Operating System: Microsoft - Windows 10 Home OEM 64-bit ($94.89 @ OutletPC)
Case Fan: be quiet! - SilentWings 3 PWM 50.5 CFM 120mm Fan ($17.99 @ SuperBiiz)
Total: $911.12
Prices include shipping, taxes, and discounts when available
*Lowest price parts chosen from parametric criteria
Generated by PCPartPicker 2018-09-11 11:33 EDT-0400