If money is tight, then the cheapest solution would probably then be an nvidia GPU with NVENC support, since you wouldn’t have to upgrade anything to make it work (assuming it fits in your case of course) . I believe I managed to grab my 1050TI for about $100USD, and it supports many, many streams, especially with the NVENC unlock driver patch, and my cpu sits pretty at maybe 10% usage tops.
Both are good ideas! A newer Gen cpu is never a bad thing in this hobby. Either method will do you good, so if you’d rather have the extra cpu horsepower of a newer cpu, you should go for it!
I have no opinion or knowledge of which Intel cpu is best for you besides “bigger number = more better” so hopefully the other responses with advice are accurate. Good luck!