norabombay suggested that I go to www.headscarves.com.
I went there, and bought a babushka - in this case, a triangular scarf made with extralong ties in two corners. I chose one made of lightweight denim layered with CoolMax. All of their headscarves are layered in CoolMax. I got the large version to make sure it covered all my hair. It arrived yesterday.
I wore it today. It tied on easily and tightly, and while it didn't drape the way I wanted, I think a washing should take care of that. I tucked the point under the knot, and my hair was completely covered, but because it is a scarf, there was ventilation, and because the underlayer is CoolMax, it absorbed my sweat very well. It's designed for this sort of thing - total coverage and being active. I barely had to adjust it and it was never in any danger of falling off.
I consider it a success.
Thank you again.