How important is the GRE for PhD engineering applications?

I have a high GPA in my major but suck at these types of tests, how important is it to get above average on the GRE? Also, do they care about the sections besides the quantitative (math) ?

It depends on the school. Usually as long as you meet or exceed some minimum score it doesn’t matter. It usually, therefore, can’t really help you but can hurt you.