Since you are mostly trying to avoid a doctor's visit and aren't opposed to medicinal treatment, just call the doctor who prescribed the inhaler and tell them you have thrush and give them your pharmacy phone #. Since this is a common issue, hopefully he/she will just call in a prescription for you. I work in an allergy/asthma office and we occasionally have this come up.
I teach our patients to at least rinse, but ideally to brush their teeth and tongue after using Advair, Symbicort, or any other inhaled steroid.