>> I prefer NOT to use RU!
Arlene, is it just RoundUp that you're down on? There are many older, traditional herbicides, but they tend to be more toxic or more persistent, or both. So I would expect them to be at least as hard on the flowers surrounded by red sorrel as RU.
Here are some suggestions from NC University, but RoundUp (glyphosate) is high on their list:
http://www.turffiles.ncsu.edu/...
Are the flowers spaced far enough apart that a weed torch could be used to knock back a lot of the red sorrel?