I think it's important to remember that nobody had the equipment to send a telegram in their homes. Drunk phone calls and drunk texts are possible because we can simply pick up the phone and dial or type it out, but a drunk telegram just isn't possible. A person could be drunk when giving the message to the telegraph operator, sure, but would the operator send a message that didn't make sense? Probably not. This article about some Civil War telegrams points out that the operators did their own editorializing. I've been trying to find an article I had read some years ago about Lincoln's assassination and the telegram that was sent to California to inform the press about it but I can't find it. The article compared the telegram and the subsequent article written from the information. The telegram contained the bare bones information and any typos or, in your question, drunken stupidity, would have rendered the telegram useless. Also, telegrams were charged by the word so again, a decent telegraph operator would probably discuss the contents with the drunk customer and give them advice on the best way to phrase their message. All this to say that even if someone showed up drunk, with the operator intermediary, their drunken message would have been heavily edited.

Here's an old thread you might be interested in: https://www.reddit.com/r/AskHistorians/comments/2su41j/is_there_any_evidence_of_a_drunk_telegram_similar/

There are a couple of mentions in that thread.