Budget inn along the canal of Amanohashidate

The inn is called Amanohashidate-so. 
The best thing about this inn is, it is located by the sea and reasonable price. There are Japanese style room and Western style room available.

Access: about 10mins walk from Amanohashidate station

You can book through booking.com

Japanese style inn in Miyazu

The inn is called Takashimaya, Ryokan. The inn has Onsen. Some of the rooms are renovated in 2010.
I can recommend their meals and better to book dinner and breakfast.

Access: 
about 10mins walk from Miyazu station
If you take an express bus from Osaka or Kyoto, you can get off at Kisenba-Minato-kan(one stop after Miyazu station) and walk about 1 min.

Booking is only possible in Japanese website, such as Rakuten Travel....

Japanese traditional inn in Miyazu

The inn is called Charoku Honkan. 

It is a Ryokan with Onsen(hot spring). 

The inn founded around 1716-1735 and the building is registered as a tangible cultural property of Japan. 

Better to book with dinner and breakfast because there are not many places for dinner(the owner told me that they also can serve western style breakfast, if the guests wish and when they are not busy).

Access: 

10mins walk from Miyazu station. 

If you are taking the express bus from Kyoto or Osaka, you can get off at Kisenba-minato-kan bus stop(one stop after the Miyazu station) and walk about 3mins.

You can book from booking.com or Rakuten Travel and so on.