Does Exit Counseling Affect Grace Period?

Hi,

Recently, I finished exit counseling but I'm worried that it started my six month grace period already. This is because I fear that it "marks" when your graduation is. I'm not sure if this is true or not but my graduation is far from today and I don't want to start paying for a loan that's not supposed to be due until a little less than a year. Should I contact customer service? Did I majorly screw up on this?