I'm not a doctor or any other medical person, but yes, I believe you can have asthma and not have low oxygen in your blood. My allergist says I have asthma and my oxygen is 100% - because I've never smoked.
Your doctor must be doing something if he's given you symbicort and a rescue inhaler. If you feel he is not treating you enough you can try to get a second opinion from another doctor.
Is your doctor a family doctor or a pulmonologist (lung doctor) or an asthma and allergy doctor? They all treat asthma.
I hope someone else who knows more than I do about asthma will come along with more help.