May starts out the play on the wagon—from alcohol, and from Eddie. Of course, she falls off the wagon in both respects.
"Decided to jump off the wagon, huh?" (216)
It doesn't take May too very long to get back to a couple of bad habits: drinking and engaging in a nasty, codependent, incestuous relationship with Eddie.
MAY: We've been drinking a little bit, Martin.
EDDIE: She hasn't touched a drop. (354-355)
Apparently as part of his ongoing attempt to paint May as a liar, Eddie actually denies that she's been drinking, even though she's more than willing to cop to having fallen off the wagon.
"Could you hand me that bottle, please?" (384)
Eddie is clearly a big fan of "the bottle" in this play, and here he and his bestie for the night (i.e., tequila) are about to connect again.
"No, no. Don't go, Martin. Don't go. You'll just get all blue and lonely out there in the black night. I know. I've wandered around likely like that myself. Awful. Just eats away at ya'. (he puts his arm around MARTIN's shoulder and leads him to table down left) Now just come on over here and sit down and we'll have us a little drink. Okay?" (453)
When he's not trying to show Martin how masculine he is or paint May as a pathological liar, he tries to get Martin to pal around and even drink with him. Weird . . .