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You are right that Flovent is a glucocorticoid (steroid). So is Veramyst. While some of the fluticasone (the medication in both Flovent and Veramyst) is absorbed systemically, it is a very tiny amount and should not cause serious mood problems. Most asthmatics are on Flovent to prevent asthmatic flare-ups. Your provider could try tinkering with the dosage of these two medications and see if that helps.
It is also usual to add Benadryl on allergy shot days. While Zyrtec can help every day, the Benadryl really helps when your immune system is being attacked with allergens.
Maxair is a "rescue" inhaler - designed for just what you are using it for. It is used when you are feeling short of breath with chest tightness. These symptoms are supposed to be kept in check with the Singulaire and Flovent, but sometimes you just need that pirbuterol to help open airways.
I checked cross-reactions among all the drugs you mentioned and found no interactions.
Going back to once a week for the allergy shots is a good plan. They may even want to dilute further if that isn't enough.
Another possibility is increasing the dosage of psychoactive drugs to control the moods. Usually ENTs are not comfortable with that though, and would prefer that you see a psychiatrist for medication management.
Have you tried upping your level of exercise. In psychoneuroimmunolgy clinical studies that has been found to help with depression associated with a number of chronic illnesses. The ones I am most familiar with involved patients with HIV and cancer.
I would suggest a call to your local university-affiliated hospital to see if telemedicine is available and what specialties are available by that route, and if it would be possible to set up with a provider in your area who would be the eyes and ears for the telemedicine specialist provider.
Please let me know if you have questions.