The key to getting super drunk is using the casks as fast as you can. The best location, as already mentioned, is the one with the 5 casks nearby. You need to go from cask to cask using them as fast as you can until you see the message "Superdrunk" then you will be able to smash the nails. A good tip is if your Superdrunk and far away from the nails, you can stop and just wait until your less drunk then go near the cask closest to the nails which is pictured south in the photo below, and use it again to become Superdrunk. Just don't wait too long or else you need to use a lot of casks again. Also, as you noticed, the nails and casks have a cooldown.
The best method is always going to be with a partner: Get or pay someone to push you as you aren't moving all over the place when being pushed. Usually, you will ask someone for help then help them in return to "pay" them.
Best method solo IMO (My world is optional PVP so magic walls or wild growths don't work): "Exeta Res" or challenge the Celebrating Orgers. This method is only good if players in your world are nice and if you have someone to challenge the Orgers. Get crates or parcels (the Orgers cannot walk over objects) and trap the Orgers like in the image below. You could probably get creative with the maze:
Credits to: Reddit for the idea!