After upgrading to 13.04 - .sh files can't be opened

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This is not a question! This is a regression BUG that happens after upgrading from UBUNTU 12.10 to 13.04.

As I've already explained, mouse right click on .sh files doesn't have an OPEN option, only OPEN with text editor.

Although the files do have EXECUTE permission correctly, the .sh files can only be opened with text editors, but the can't run!

I am assuming you are talking about opening with the nautilus file manager. Nautilus is the default and has a icon that looks like a filing cabinet in raring.

To run a shell file from the file manager you need to

1. Make sure the file is executable; sounds like you have done that already

2. You need to change the default action in the file manager

From the "Files" menu select "Preferences" then in the window that opens the "Behaviour" tab. From here you can change the Option for "Executable text files" to either "Run" or "Ask" as you prefer. This setting is remembered so you only need to do this once.

OK - but I've already done that while I was running 12.10 so there's no need to force the user to do that again after upgrading to 13.04