I am integrating the Google Plus in my Android app. I have created the project in the Google API Console. I created the OAuth Client ID and I doubled check the package name
Do you get any output in logcat to go with the error? Try enabling verbose logging if you haven't (see https://developers.google.com/+/mobile/android/getting-started#frequently_asked_questions)
adb shell setprop log.tag.GooglePlusPlatform VERBOSE
Edit: Thanks for adding the logs. The GLS error is the interesting one there:
I/GLSUser(1699): GLS error: INVALID_SCOPE xyz@gmail.com oauth2:PLUS_LOGIN
Can you check that you have enabled the Google+ API in your API console project, and can you also try removing the setScopes line entirely (PLUS_LOGIN will be the default, so that seems the easiest way to test if anything interesting is going on).
I'm not sure off the top of my head whether PLUS_LOGIN being in their is indicative of a replace not happening, or its just the logging looks like that, but remove the line should remove it from the equation.
EDIT - based on your updated dump the scope now looks good but there is an invalid client Id error in there. Make sure there aren't any trailing spaces around your sha 1 in the client Id and that the package name exactly matches. Also try clearing the cache in google play service (clear data from the menu in the google settings app).