Don't think you can close Stickies and then delete one. What you describe works for me. I created a new one, deleted it as you did, then closed. When I went back in, that note was not there. I believe that the notes you create are stored in a StickiesDatabase file. Settings for Stickies are in the com.apple.stickies.plist file, and perhaps that is where you can solve this problem. Since the actual entries are in that database file, getting rid of corrupted settings should not delete your notes.
First, restart the computer. If that doesn't solve it (or you've already tried that) then go after that plist file. It is in your User's Library folder, that is: HD/Users/YourUserName/Library/Preferences. Find that file and drag it to the Desktop. Now restart Stickies. Your Mac will make a new plist file set to defaults. Does that do the trick?