Where does the OSMO save the stitched pano? I only see the folders on the SD card with each image but not the stitched file. On my iPhone the stitched file is saved in my photos but cannot find them on the SD card.

Hi, they are saved to your mobile devices, and not on SD card.  Because the OSMO needs to use the processing power of the mobile phone to stitch, so it first transfers the photos to the phone and leaves the generated photos there.
Also, the stitching function in OSMO is intended for generating panos for quick sharing. For better quality photos, it is recommended to use 3rd party apps on PC to process,   as: https://www.dermandar.com/create/   
http://hugin.sourceforge.net/