Will amoxcillin get rid of mucus in sinus cavities? On day 3 of med, still feeling dizzy and bouncy when I tilt head down.

Steroid inhaler. An antibiotic might be appropriate if one has a yellow or green nasal discharge, especially if associated with painful sinuses and/or fever. If the discharge is clear, sinuses are pain-free, and there is no fever, a more appropriate medication might be an inhaled steroid such as Flonase (1-2 inhalations every 12 hours for7-10 days). If symptoms persist, see your md or an ear, nose and throat doc.
Not for you? Amoxicillin is commonly prescribed for bacteria originated sinus issues but your infection may be resistant to the antibiotic or maybe viral. If it doesn't start to resolve in a few days, seek out an ENT or your gp for more help. The generally know what "bugs" are circulating in your area and might have a better handle on the appropriate treatment.
Was it examined? Was Amoxicillin prescribed by a doctor to treat your sinus? If you are self-medicating with antibiotics on hand, it may not be the right type of drug, so see a medical doctor. If your doctor prescribed amoxicillin, please contact your doctor to communicate your lingering symptoms. You may be advised to continue it, or switched to another kind if your response is not good.