Jack Del Rio

Cops say they were able to catch up to the former NFL head coach on a street near the crash site -- and in a police report, officers wrote he reeked of booze.

Jack Del Rio had a miserable night in handcuffs following his OWI arrest last month ... TMZ Sports has obtained police video showing he begged an officer to let him "take a leak" -- before he repeatedly screamed and moaned in pain while in the back of a squad car.