You could have hardware problems. Is this the same machine you use to record YouTube or other on-line sound tracks? Those settings will profoundly affect the sound quality.
You are experiencing both ends of product development. You are using Skype for this job precisely because you know Skype is going to connect correctly and establish a clear, reliable conversation pretty much no matter what. They do that by taking over your computer. Skype works no matter how screwed up your computer is.
The down side of that reliability is no options. There are only two known ways to reliably do production with Skype: Use a software package that knows what Skype is such as Pamela Professional or Pamela Business. Both of those will provide a split stereo sound file with you on one side the the guest on the other. Getting from there to what you want is a snap.
http://www.pamela.biz/en/products/
Other people make Skype capture software.
https://forum.audacityteam.org/t/help-recording-with-a-mic-laptop-and-headphones/35068/6
The other way is two computers. that’s how I did it.
And that’s how most “grownup” productions do it. This is a scene from Pando Podcast.
All that and it’s still possible you have your microphone set up wrong. Look at the Audacity Device Toolbar and make sure you’re recording from your microphone specifically and nothing else. It’s also possible to get your local microphone set up and Skype will stop working.
Isn’t this fun?
Koz