So to clarify, if drivers aren't the issue (using Windows 10) AMD R9 380 or 380X seems like the best choice for $250-300 price point?
The 4 best cards in the $250-300 range are R9 290/290X/390 and 970.
The easiest is to work backwards:
GTX960 2GB/R9 285/GTX960 2GB should be discounted as 2GB is not a good choice.
GTX960 4GB is
$165. R9 380 4GB is
$180.
GTX970 is
$280 but comes with a choice of AC Syndicate or R6S.
XFX R9 290 is $240 and has lifetime warranty.
PowerColor R9 390 is
$280.
With your CPU and the free AAA game, I'd pick MSI Gaming 970 over the R9 390.
I'd eliminate the GTX960 because in GPU limited games,
its performance is abysmal compared to the R9 290. Honestly, even with your i3, I am now leaning towards not recommending even a $165 GTX960 against a $240 R9 290, even with your Core i3.
DX12 benchmarks - the only ones we have - show GTX960
bombing. If you intend to keep the card for a while and play DX12 games, do you want to take that risk? Plus R9 290 has 64 ROPs. Will 960 perform better in other DX12 games? Maybe but why take that chance if your budget is $250?
The second reason is some games aren't actually very CPU limited. So what happens in those games? Well, the GTX960 completely falls apart in GPU demanding titles where CPU is less relevant. Star Citizen is a prime example of that. The performance of a 960 in SC is horrendous compared to an R9 290X (~ after-market 290), even when paired with a low end AMD CPU like the 860K:
or this
or this
That means even though you will be CPU limited in some games, in situations where you are going to be more GPU limited, 960 shows its true colors - aka it's not a very good graphics card. If you are CPU limited, with a 290 you can at least increase MSAA/AA, pile on mods, enable VSR and improve your IQ. With a GTX960 you have no such options since the GPU is underpowered.
I think for you, I'd look into a used after-market R9 290/290X, a $240 R9 290 or a $280 GTX970 or wait for a deal on a 290/390/970. Sometimes EVGA b-stock has a $240 GTX970.
I know on paper GTX960 sounds like a good fit with the i3 but every time I come to terms that 960 may be an OK card, I see it just bombing in newer benchmarks like Star Citizen which makes me want to not recommend it once again.
Then again, I can't deny that if you have an i3 2120 or something along those lines, you will get bottlenecked in A LOT of games which means if you go R9 290/970, you are not going to get max performance out of those cards either. You also may need to bump your RAM from 4GB to 8GB too. Maybe buying a used after-market 290 and another 4GB DDR3 stick could be the best choice for you as spending upwards of $300 on a 970 may not be the brightest idea either.