Yes, it could take a week or so to see your symptoms go away. It may happen before that, but don't be surprised if it takes that long.
I don't know what they told you when you tested, but you shouldn't have sex for at least 7 days, or while you still have symptoms, whichever is longer. Make sure all your partners are notified, as leaving these untreated can cause lifelong problems for both men and women.
The azithromycin will cure chlamydia, and may cure gonorrhea, but there is some resistance to it. The shot will cure the gonorrhea.
Have you been tested for Mycoplasma Genitalium. Wait 3 weeks after antibiotics and use first void urine only. MyLabbox can do a home test.