hmmm, that's a very close call. For RE: 2 the min CPU is an i5 4460, which is slightly better for everything than your current CPU.
The 960 might cut it with a better CPU, but you will have stutteringI expect. Perhaps not so much at low settings, but anything higher will impact it.
You are right on the threshold of playable experience, that's enjoyable for that game, and your current rig.
I know it's so tempting to just get the GTX1060 and weigh in. But you may be disappointing. Some games most certainly you will see an improvement. But for some not so much at all.
I'd recommend waiting a bit saving a some extra dosh, and getting something that wll give you a solid base (upgrade-able) and the potential for a lot more performance than you currently have.
If you absolutely must have the 1060, then just be prepared, that in some games you might not be overwhelmed by the expected increase in performance versus the actual increase.
Good luck with the choice
Just to whet your taste buds, you could do something like this:
PCPartPicker part list /
Price breakdown by merchant
CPU: AMD - Ryzen 5 2600 3.4 GHz 6-Core Processor ($164.89 @ OutletPC)
Motherboard: ASRock - B450 Pro4 ATX AM4 Motherboard ($64.99 @ Newegg)
Memory: Team - Vulcan 8 GB (2 x 4 GB) DDR4-3000 Memory ($59.99 @ Newegg)
Total: $289.87
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2019-02-07 14:23 EST-0500
Along with your GTX960/HD/SSD/PSU you would have a PC capable of playing those games you mentioned very nicely at pretty much med-high settings at 1080p.
Then, all you would need down the line (maybe 6 months later) is a GTX1660ti/RTX2060/RX590 for a really good gaming experience at high/ultra settings at 1080p.
Making good choices now, will save you money. Getting a GTX1060 with your current system, will cost you money.