I finished my azithromycin on tuesday. How long will it take before my sinuses drain?

Varies. This med is given for a short period but stays in your system killing germs for at least a week after you stop taking it.If the antibiotic is working you should have reduction in pain or pressure as a sign of progress. Clearing of the sinuses can take a few weeks. If you have had no improvement by now contact the doc and discuss your remaining symptoms. Another med may be required.
Sinusitis. Azithromycin is an antibiotic which may inhibit or kill bacteria causing sinusitis, but the inflammation may create congestion which requires a decongestant or irrigation to implement drainage. Two different issues.
Azithromycin. The medication remains active in your body 5 or more days after your last does. Depending on the severity of your sinus issues, it may take up to the full 5 days after your last pill before you see improvement. That said, if you are not at least seeing some improvement, contact your prescriber. Sinus infections are notoriously difficult to treat sometimes!