It's gonna be difficult to find AM3+ on the market still, a lot of stores stopped stocking their processors over a year ago. Otherwise if you can still find some, any of the FX 8 core would help but keep in mind you might actually lose single core performance depending on the processor you currently have.
I have an 8350 and it has done fine for gaming & recording with OBS, when I did stream using OBS I was able to stream it at really good quality, I just set it to the maximum I could for my bandwidth even thought Twitch limits the quality. It has only started showing its age in the last half of last year though.
At this stage, my advice would be to avoid spending any further money since the AM3+ socket is so old now. It's outdated & no longer supported. If you can hold out for an AM4+ then wait. Otherwise, you'd be better off just upgrading the system entirely. The Ryzen chipsets are doing very well at the moment & AMD plan to support them for the remaining 3-4 years.
1 other thing you might want to look into is changing OBS to use your GPU instead of CPU. This is possibly why I also had no issues streaming as I had originally change it from CPU to GPU.