*I am not a teacher. This is only my personal opinion.*
Some people drive even after they have drunk.
Suggestions- have drunk alcohol, have been drinking (alcohol), are drunk.
If a person has drunk
Suggestions- has been drinking, is drunk.
Accidents often happen on rainy days when a vehicle skips
Did you mean to use "skid" here? Skid is used quite often in this paragraph, you might want to edit it a bit. Perhaps change the last phrase from "will skid and collide with one another" to simply "will collide with one another".
Drivers should change the tyres of their vehicles [when] they are bald as bald tyres do not have a firm grip of the roads.
Suggestions- a firm grip ON the roads or do not firmly grip the road. I'm not certain that "of" would be considered wrong though.