I was able to get a Shiny Regidrago by soft resetting. I made sure I wasn't catching the same one by catching one, checking its stats, and soft resetting. Then do that again. Use a Master Ball if you have one.
Phew, I'm glad I didn't to those soft resets for nothing then. Thanks for the info~SRs absolutely do work. I got Regice right away, but unfortunately, it took me nearly 9 unlucky days (~65-70 hours) of SRs for Regirock.
I'm currently at 3 days for Registeel. A lot of people have been going over charm odds, so some are believing the Regis are standard odds, but that isn't confirmed. A lot of people are just possibly getting unlucky.
Still no shiny Regidrago, but I had some legendary hunts taking weeks in Ultra Sun, so this is indeed pretty normal if you're not being on the lucky side for a particular hunt.