I ended up bringing all the information together and now have a solution.
This takes into consideration the following facts:
firstScript = variable name that points to a script called scriptNumberOne.sh
scriptNumberOne.sh = the script that I have embedded into my application to run
ButtonHandlerRunScript_ = the name of the Received Action in Xcode
pathToResources = variable that points to the internal Resources folder of my application, regardless of it's current location
Using this information, below is a copy of a vanilla AppDelegate.applescript
in my AppleScriptObjC Xcode project:
script AppDelegate
property parent : class "NSObject"
property pathToResources : "NSString"
on applicationWillFinishLaunching_(aNotification)
set pathToResources to (current application's class "NSBundle"'s mainBundle()'s resourcePath()) as string
end applicationWillFinishLaunching_
on ButtonHandlerRunScript_(sender)
set firstScript to pathToResources & "/scriptNumberOne.sh"
do shell script firstScript
end ButtonHandlerRunScript_
on applicationShouldTerminate_(sender)
-- Insert code here to do any housekeeping before your application quits
return current application's NSTerminateNow
end applicationShouldTerminate_
end script