If you ever find yourself in a position where you're drunk and have to deliver a dead body across the state, I'd suggest staying at home. That's because drinking and driving is stupid, and the last thing we need is another dead body to deal with.
That being said, if you're a drunkard and insist on risking it, I'd suggest using a hearse or ambulance. They are not only effective and traditional forms of corpse transportation, but the odds of a cop pulling you over for DUI are slim to none. If they see you swerve, they'll just assume the body came to life, is attacking you, and leave you alone.