You will get an OWI if you ride a horse drunk.
Source: I know a guy who trained his horse to ride from the bar to his house on its own. Cops still pulled him over because he was sleeping on the horse.
I’m pretty sure it depends on the state and whether or not that state considers a horse to be a vehicle/device. Alabama, for example, I believe does not consider a horse to be either, while I think California does. There’s this story that sometimes gets submitted to TIL-type communities where a man from Louisiana was decided to be ineligible for a DUI charge after doing exactly that, but he was still given a court summons for “disturbing the peace by intoxication”.
In sweden there were some cases where people lost their driving license because they … Walked home drunk so yes it do depends a lot. Guess drunk horse riding there is not legal.
I think you may be talking about endangering traffic, not just walking while drunk.