CPU cores matter for streaming. It is tough to stream on an i5. The extra threads of an i7 will help a lot. I agree with above and would go with an i7 4790. I would look for a used one on ebay. Should be able to drop it in your current motherboard.
If you better fps and increased graphic settings, then you should look at the GPU. GPU prices are really high right now, the best you could afford would be the 1060 3gb. But I would suggest going over your budget to $350 and go for the 1060 6gb as it will be a really nice bump from your 960.