As I drink more beverages, my drunkenness slowly increases. Does this have any effect on my abilities, negative or otherwise? I haven't noticed any impairment at all until I'm fall-down-drunk at which point I can't do anything at all.
It has no effect at all until you are falling down drunk.
You cannot drink anymore booze once you are drunk, on St. Sneaky Pete's Day there are special drunken stupor adventures when you reach 26+ drunkenness, you can drink green beer on that day to get 10 above your normal limit.
There is also a Trophy for going to the Trophy hut in Bordertown with 1000+ drunkenness. The most efficient way to get this drunk is using the Dungeon of Doom potion that gives you drunkenness. The specific potion that has this effect is different for every player and between ascensions.