OK. Let's try this...copy & paste this command at the prompt and press return.
defaults write -app 'TextEdit' ApplePersistence -bool no
Now restart.
If for some reason this doesn't work (but I bet it will) In System Preferences > General there is a hard-to-find checkbox under "Number of recent items" you can turn off. (Also, you can hold the shift key when logging in to disable resume on a one time basis.)
Another way to accomplish this is through Finder by going to ~/Library/Saved Application State/TextEdit and delete that file…or any other app, like Safari.
(note that the User Library is hidden - so go to your Finder "Go" menu hold the option key to choose "Library") The go to "Saved Application State" folder and delete the files for TextEdit and Safari.
Or to turn off Resume globally...
chflags uchg ~/Library/"Saved Application State"
Press return.
The reverse should you need it is
chflags nouchg ~/Library/"Saved Application State"
Let us know.