I stream on Twitch every now and then and always have noise in the background, but the noise does not come from my headset. I unplugged it just to see if the noise went away. I tried using OBS and AMD Streaming. The same result for both.
Buy a new microphone
The microphone isn't the problem. As already written, I tried it without and the noise was still there.
Buy a new PC
I think that would be too early. Have only had it for a year