and I actually have something cooking that might be exactly what you're looking for. Keep an eye out!
In the mean time, here's a system for drinking contests I came up with a while ago:
Every player rolls a d20 (
/roll 20) with each drink. For the first drink, any roll above a 1 lets you keep drinking comfortably. However, after each roll, the number you have to beat goes up by 1. So for each player's second drink, they need to roll above a 2. Third drink, above a 3, and so on. If you ever fail to roll above the number, you're too sick to continue, either passing out or throwing up, and are probably out of the contest.
This creates a nice sense of rising tension, perfect for a competition!
If you want to account for differences in constitution, you might let each character add one fifth of their
Physical Stat as a bonus to their rolls. Alternatively, because we're playing with virtual dice, you could allow that bonus to adjust the size of the die itself. So someone with 30 Physical stat is rolling a d26 (/roll 26) instead of a d20, since 20 + 30/5 = 26.
While the latter method guarantees no one gets any "safe" drinks, the former method might be more intuitive to those who are used to other d20 games.