Other mistake: When Rex shakes the table causing the baby monitor to fall, the batteries fall out and we see the open battery compartment with markings for positive (+) and negative (-), and space for two AA batteries. Woody jumps off the bed and places the two batteries inside the compartment, but before he closes the compartment note the position of the battery closest to the top of the monitor. The positive side is actually where the negative side should be so the monitor should not work, but the monitor turns on just in time to hear the soldier's warning. (00:13:15)
Answer: There wasn't anything for him to get drunk on. He was just wallowing in self pity and frankly, his world had just fallen apart, so it's understandable that he'd act a little loopy as a reaction.
Garlonuss