Before adding an additional Publish profile in Visual Studio, everything worked great. When I right-clicked on the relevant project and selected 'Publish', I was able to deploy just that single VS project to my Azure app service.
After adding a 2nd Publish profile which deploys to an additional app service Staging slot, right-clicking on the relevant project deploys all VS projects in the solution with either profile.
I found this posting, but not sure where/how to make those changes or if it's the right thing to do.