Currently, I cannot find a way to export RAW photos I am editing in Photoshop Express back as a RAW format file (DNG). Is there a reason for this? I'd love to have the ability to save the file as a RAW file again.
An edited RAW file is no longer a Camera Raw file -- it has been through extensive processing to make the RAW file editable and look like you expect (demosaicing, noise removal, lens correction, etc.).
So no, it does not make sense to save edits back to a Camera Raw format, and they should be saved in a normal image format (TIFF, JPEG, PSD, etc.).