Welcome to the forum.
There are two places that one must choose Audio in Export/Share settings: initially, with checking Export Audio & Export Video, and then with the Multiplexing Tab, where the method of combining the Audio & Video into one file is chosen. This is the point usually missed. What is your Multiplexing Tab set to?
What happens if, when you chose Export Audio & Export Video, and set the Multiplexing to "DVD," or anything but "None," and then Export/Share to a folder on your HDD? Do you get muxed (Multiplexed) Audio & Video in one file then?
You could reinstall the sound card driver to see if sound encoding is the real issue. When you play it back, it is only decode. Only when you share/export does encode come into play. Drivers could have a say in this. Otherwise, a reinstall could help because the other components required for encode (codecs) are installed when the app is installed.
You can also delete the media cache (Go to Start->run type in the following within quotes "%appdata%\Adobe\Common\" and delete everything there.) Restart Premiere Elements, reopen the project and let the conforming happen. Export after this and see if it helps.
If your audio is in either Soundtrack or Narration move it to a higher Audio track (e.g. Audio 2). For many that resolves sound issues.
Insanity is hereditary, you get it from your children
Thanks for the reminder. Earlier versions of PrE seem to have issues, at times, with both the Narration and the Soundtrack, fixed Audio Tracks. I have not read of those in PrE 10, but because they seemed to have been intermittent in nature, maybe they have just not bitten?
Due to the problems, that others have written of, I just ignore those two fixed Audio Tracks, out of habit.