> If a driver appears to be drunk, how is that a random stop?

They're not stopping drivers who "appear to be drunk" - that's the point. They are allowed (supposedly, IANAL) to set up a checkpoint to stop every car (effectively a random stop) that passes by to see if the driver is intoxicated, or check to see that they are not in the country illegally.

They're not allowed to check for other crimes without probable cause though. They can't set up a check point to say drug possession or tax evasion, though if you are at a DUI checkpoint I believe you may somehow give them probable cause to check for those things.