What Can Cause Dizziness And Ringing In The Ears?
Ear infection wouldn't explain all the symptoms.
Detailed Answer:
I read your question carefully and I understand your concern.
Some of the symptoms like headache, dizziness and buzzing (if on one side) might appear in the setting of an inner ear infection. However such an infection wouldn't justify that passing out feeling and the hand shaking, even if you had an inner ear infection shouldn't be responsible for that. The same would go for the sinusitis suggested by your doctor though, so not sure I agree with him either. In any case some cheap routine blood tests like a blood count and metabolic panel might be useful to search for changes indicating infections, anemia (which could produce those symptoms) or metabolic abnormalities. If your doctor strongly feels it's related to sinuses sinus imaging may be done as well.
If tests turn out normal and there are no other symptoms or findings in physical exam then I would also consider the possibility of anxiety. It would justify the hand shaking, dizziness and passing out feeling which are common with anxiety. It is also commonly associated with tension type headache which manifests with the headache and pressure feeling you describe.
I remain at your disposal for other questions.