50% of women with chlamydia have no symptoms, so that's no reason to not believe your diagnosis. Chlamydia can cause cervicitis, so almost definitely that's what you have going on.
As to the antibiotics, 1g of Zithromax is a very effective treatment (if that is the dosage you have). More, like 2g, would be even better. There should be no reason to need to take the doxycycline at the same time.
The best bet is to take your zithromax and then be retested later to be sure you're rid of the chlamydia. You should wait at least a couple weeks after taking your antibiotics for a more accurate result. And don't have unprotected sex in the meantime!