Thank you, Boyan.
I am experiencing the same thing as you. The connection string does not exist in the ExecutionContext.DataContexts collection.
Fortunately, I am able to get the BaseUrl from Manager.Settings.Web.BaseUrl, and that helps a great deal. I am now storing my connection strings in a utility class as noted in my original post, and I can use the current BaseUrl to determine the correct connection string I need at runtime. This accomplishes what I need.
Thanks again for your help!