I always have problems with this. This is part of the reason I hate dialogue. I somehow get myself into a corner where some drunk is talking to the main character and I'm stuck trying to figure how to write out a drunk's slurred speech. I don't know why, but I get negative reviews on it and it seems as though it's just because the person reviewing doesn't like that I use dialect in the dialogue. I don't understand. Mark Twain used it extensively and his books are classics, and it's not as though my stories are riddled with dialect and dropped consonants outside of dialogue. Is there some rule that it's not acceptable for writers to use dialect in their stories? It doesn't seem right. There are tons of people who don't speak properly. Why can't I incorporate that into the dialogue of my stories? Not every character is an English major.
The trouble is, not everyone will understand the dialect, or the phonetic way you attempt to spell it. You need to suggest the speech without going into full-blown transliteration of how people are talking. It would be extremely irksome to read a whole scene of dialogue written out, e.g. with "sh" instead of "s" for a character who lisps or is slurring his words because he is drunk. Maybe that's why you got bad feedback. Just using "sh" once or twice is enough, even if in fact all of the "s"s would be slurred.
Don't confuse dialect and accent/phonetics. Dialect is words or phrases characteristic of a region and/or social niche. Accent and phonetics are the rendering of ordinary words so they sound different to a listener. It's best to avoid writing accents and phonetics. They bog down reading. Normal reading is by recognition of words as a single unit. Phonetic rendering requires the reader to sound out each syllable It's an unnatural activity, and makes reading like jogging from a firm cinder track onto a stretch of ankle-deep muck. Sounding out drunken slurring of speech falls in the latter category. If you must do it, keep it brief.
^^I agree with Madhoca and Cogito. You don't need to write every word in dialect - the odd word here and there will give an essence to the dialect.
A classic case of tell, don't show. Don't try to show us with phonetic spellings, just tell us that his speech was slurred. Or if that's too direct for your sensibilities, tell us that somebody has trouble understanding his slurred speech.
And if you're more interested in 'showing' a character is drunk, rather than telling, for all the above information is spot on.. you can always do so by the things the 'drunk' is talking about. Your POV character will obviously be coherent and logical, though your 'drunk' may swerve through subject matter is if he were driving blind. Hemingway has great scenes where characters are getting drunk. He only mentions the fact that they are sipping wine, or drinking something, and the conversation itself reflects the level of intoxication, not phonetics.
I agree that drunkenly slurred speech isn't dialect, but accent. (Perhaps temporary accent, but accent.) I'm fine with dialect being shown in stories, through word choice and order, but accent just doesn't work for me, even in the works of the finest of writers.