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I"ve actually just written a scene in a novel, in which a recent stroke victim slurs her speech. But I don't try for "realism" in this, I write her dialogue straight; but the characters she is talking to struggle with what she is saying.

At the beginning, she apologizes for sounding like she's just been to the dentist.

A few lines in she says, "I was trying to go to the store." I write it like that. 

But another character says, "I'm sorry, you were ... oh! Trying to go to the store?"

The stroke victim nods, and says "Yes."

In another sentence, the stroke victim can hear herself, and tries twice to say a word, then finds a synonym to say instead. 

Try talking while holding your tongue still, firmly at the bottom of your mouth. Say the lines out loud. It will help you figure out, realistically, what is understandable and what is not.

I personally don't like "phonetic" slurring of speech, it always breaks immersion for me to have to figure out what text is trying to say. I'm quite sensitive to typos for the same reason. I don't catch every one of them, but when I do they stop me in my tracks. So it may just be a quirk of my personality, but I'd suggest don't try to show what the slurred speech sounds like, show the reactions and difficulty people have with slurred speech.

+1 @Caspian for the recommendation on pacing, stalls, confusion and repetition, in portraying drunkenness.