Dictionary meanings- Dogal means halter of any kind (could be a rope). It also means a hangman's rope in one dictionary that I have. In some usages, it means to be in a tight spot.
The Museum gives no supporting information in their short definition.
Cowboy Museum

The National Cowboy Museum is a large, modern, attractive facility on a beautiful piece of property, with big cattle and oil money behind it. The staff, including research directors, is well-paid. I would like to see the exhibit on Vaquero rawhide braiding currently showing. Their magazine, "Persimmon Hill," is glossy, full color and professional, history, art, personages, - my daughter has a few, one I looked at recently features the history of the Hamley saddle.

At this website, don't miss the large collection of photographs of cowboys, 1880s- early 1900s!