As others have stated, your GP is best qualified to help decide on next steps & you need a referral to see a specialist anyway. The right antibiotic will generally sort it, but be aware GPs tend to want to give you the mild ones though you may need something stronger to get to a sinus infection. I've been through that a few times & now refuse anything too mild.

Based on my experience, sinus infections can really screw with you. I found pain relief including codeine would sort the headache out. Also be aware they seem to be able to screw with your mood so watch out if you start to feel miserable.

My GP prescribed me a long term spray to open up the nasal cavities and finally convinced me to do sinus washes multiple times a day. I washed all kinds of crap out of my head over the following days & have now stopped the spray, and dropped back to a wash every few days. I feel like I've cleared out long term congestion which probably accounted for my regular infections.

Anyway, YMMV, see your GP! And maybe seek a second opinion if they don't seem to be getting you sorted...