Is it worth to upgrade my i7 860 processor?

I have an Intel i7 860 @ 2.80 Ghz, sapphire r9 270, 10gb ddr3, 1156 biostar motherboard, 750w psu. I want to upgrade my PC's processor. I can play fallout 4 on high settings but when I go to big places there's little input lag spikes and same on witcher 3 on medium. And even on minecraft. I play battlefield 4, and black ops 3 fine but there's still those lag spikes. So it made me think that it's my processor. Looked at the task manager specs page thing and the cpu is going over the 2.8 ghz when I play processor demanding games. But I can't get any water cooling or cpu fans because sadly the mobo is stuck to the case, stripped screw. So that means if I want to get a new cpu and mobo then I'll have to get a new case. I was thinking of the corsair spec 01 amd fx 9590 and asus fatal1ty 970 performance mobo but I've heard some bad things about the 9590 and that my i7 is better in some ways. I don't know a lot about the higher end amd and Intel cpu's. So i need help on a line of good cases, motherboards and processors that will actually be a reasonable and worth getting upgrade. I have $300 but willing to spend up to 450. Can anyone help

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In addition you will need a new windows license if you are changing motherboards - the OEM windows license is tied to the old motherboard

No, you better upgrade the graphics card than the cpu because for Fallout 4 and Witcher 3 it's the R9 270 gpu that is the bottleneck (probably for other games too).

For about $320 you can get an EVGA GTX 970 SSC ACX 2.0+ graphics card that will improve the graphics performance by roughly 100% vs the R9 270, no problem for the 750W psu.
If you do so, don't forget to uninstall the ATI driver and the ATI Catalyst CC (if present) completely prior you install the Nvidia driver.

You can also overclock the cpu up to 3.20Ghz (not higher) without the need for a new cooler.