The elf who is searching for the secret onsen is not in Begus, she is in Witton. ;-)
Also, what you have to know is that you have to interact with the border of the screen on the path in order to trigger the access to the next screen, I had a hard time to find the secret onsen because I tried to follow the path to the next screen but didn't interact with the border of the screen at first so I thought I was wrong and searched elsewhere when I was right, it was there. :-3