We're not aware of any conflict using the PRO-HD and Sling Adapter with the SlingPlayer for iPhone and Dish Remote Access apps respectively. A good first test is to connect via watch.slingbox.com and see whether the same behavior persists. If not, this might suggest that there's an issue with the apps on your phone. Uninstalling and reinstalling them should take care of that.
If the same behavior persists when using Watch, try isolating the fault by eliminating components. For example, remove the Sling Adapter and then try using the PRO-HD, or vice-versa and try using Dish Remote Access.