Can you drive with a broken shock absorber?

No, you shouldn't drive a vehicle with a damaged suspension. You cannot operate a car with any suspension component broken, regardless matter whether it is a broken coil spring or shock absorber.

How long do shocks last?

Shocks and struts often need to be replaced every 50,000 to 100,000 miles. They will age more quickly if you drive more aggressively or frequently on difficult roads. Shocks and struts don't fail all at once; rather, they lose their effectiveness over time.