Shukunegi is small, but take a look at this link:
http://shukunegi.com/en/lodging/
Its the citie's website. its listing a few options of places to stay.
You can find some other options of Ryokan near shukunegi here as well:
https://www.tripadvisor.com.au/HotelsNear-g1021355-d1977050-Shukunegi-Sado_Niigata_Prefecture_Chubu.html
I hope this helps!