It is the Yamamizuki Ryokan in Kurokawa Onsen. Can also be used by non-staying guests during daytime. I personally have not stayed at the ryokan itself, so I cannot judge the ryokan. But the bath is beautiful.

http://web.mac.com/kigouya/camps/yubara_onsen.html

As a normal practice, towel should be kept away from the spring water. mixed or not.

The twoels are kept wrapped around one's body even while in the bath.

On the otherhand, for men it is generally bad form to let your towel enter the water regardless of mixed or segregated bathing.

I never been to mixed. And all my trips to an onsen in Japan, I was always naked when enjoying an onsen. I heard that it is bad manner to have ones hair and towel get into the water.
From my experiences, Japanese people were so polite and they hardly stare. I felt very comfortable enjoying myself with the locals.

If you go barging in with everything swinging... you get the idea. As opposed to adopting a bit of a modest attitude, and just using your towel accordingly.

Actually, many mixed onsen require that women wear towels in the water. But I agree that when in doubt start off conservative, see what the locals do, and then follow suit.

If we do, how would we get there by JR from Tokyo (or just south of Tokyo)? Where would we get off?

Take the Akita Shinkansen to Tazawako Station, and then a bus to Tamagawa Onsen.

You can sort by prefecture, look at whether they have mixed baths, see pictures, and read reviews in English.