To be caught dead at (or with) means something so thoroughly humiliating and embarrassing, that you'd be ashamed even if you were dead at the time.
For example, "I wouldn't be caught dead wearing a lime green polyester leisure suit!"
In your example, it sounds like Ralph's is a grocery store. Perhaps "I" is so upper-class and posh that she normally wouldn't do anything so ordinary or plebian as go to a supermarket? I'm just guessing. (I don't think being an alcoholic has anything to do with it, given the context.)
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