Drunk driving laws are far from lax. In fact, my state puts of signs bragging about how bad you're fucked if you're caught driving drunk. Twenty thousand dollars in fines and court costs plus two years in jail, inability to legally drive for several more years plus the fact you can probably never afford insurance again, is plenty harsh enough to convince rational people not to drink and drive.

But, turns out alcoholics aren't rational people. So after they get out, they drive drunk, but this time they don't have insurance (and they're too broke to sue) so when they do cause an accident next time, the victim ends up needing to pony up the cash to pay for it.

I have several drunk family members who've been caught drunk driving multiple times. They've been to jail, and they don't have a license, but they still drive. This is not a problem you solve by making laws harsher (unless maybe you go to Sharia law levels of harsh and execute people after some number of offenses). You have to tackle alcoholism.