A drunk shit is the first shit you take after a long night of drinking, which usually is a little more stinkier than normal
Lani: Man, I had like 4 shots of patron, 3 jack and cokes and a couple of jagerbombs last night... I think I need to take my drunk shit now.
The shit you have to take the day after drinking a lot, where no amount of toilet paper will ever be enough.
"Dude, why are you walking funny?"
"I just took a drunk shit... Can I use your shower?"