I had the same problem, not for 5 cycles, but rather 5 years. In some people, including myself (but not necessarily you, no guarantees) the antibiotics kill most of the bacteria, but not all, and it just keeps growing back. The reason, according to the Mayo Clinic, is that sinus infections don't follow the usual model of tissue infection, say, like an earache. The infection can lodge itself in the upper sinuses, give you post-nasal drip indefinitely, and keep your throat soar. February is the worst month, for some reason.
Fortunately, in a desperate state like yours, I invented the Flip-Turn Sinus Flush. If you've tried regular sinus irrigation, you know it doesn't work in the upper sinuses, due to gravity. The flush defeats gravity.
http://www.medhelp.org/user_journals/show/2322
The Flip-Turn Sinus Flush is mildly risky, because you have to bend over to do it, preferably in a shower, but you can also do it outside on soft ground.
My journal will explain why the antibiotics sometimes don't work, according the the Mayo model.
I'd also recommend trying nettle leaf for your allergies. The success rate seems to run about 50%. I don't think it's herbal magic, but rather the magnesium that reduces allergic reactions.
Good luck, persistance rules.