Just to clarify -- where do you see these files? As I said before, JPEG and .jpg are identical. Could you make a screenshot?
Double-check to see that your changes were preserved in your Camera Settings. You might also check the first item, "preserve settings", although I'm not sure how that would interact with the the unchecked "Keep Normal Photo".
Also check Elements. Most photo programs create a proprietary file type from your original image. If you want to share that, you will have to export it to another format (which might be jpg by default for Elements). It does that to preserve the complete original ("non-destructive editing"), and can create a new edited version as a JPEG. If you save that file on top of the original JPEG, you will get a file warning and, if you continue, you will overwrite the original file, but still be left with two files for one photo. Photos for Mac OS and iOS does that behind the scenes, and can revert to the original after editing without having multiple files visible.