You're seeing noise for two reasons. One is the tiny sensor of your webcam, but since you don't want to buy a new one right now, you can't do anything about that. However, the sensor size is only an amplifying factor of the real problem: Not enough light. Webcams are built for video chats, quick and dirty recordings, not really for high-quality video production. You probably have not enough light in your room and haven't noticed it since the Webcam just bumps up the ISO/Exposure until it looks well-lit. The problem is that increasing the ISO basically means amplifying the sensor signal digitally, which will also amplify the noise and artifacts in your video.
I assume you don't want to spent hundreds or thousands on studio lighting. So start small. Gather a couple of lamps from your house and use them to light up your scene. Be aware that they will probably have differing color temperatures, but that can to some extent be fixed in post. For the positioning of the lamps, start with a simple 3-point lighting. Then check the results for any over- or undereposed parts of your face/body/scene. Adjust the lighting to your liking, then rinse and repeat. You may have to adjust the angle and distance of the lamps. Increasing the distance between the lamps and the subject will reduce the amount of light reaching the sensor exponentially. However, the farther away (and the bigger) your light source is, the softer the shadows get. There's no 'perfect' setup, it all depends on what you want your video to look like.