Can I relieve my allergies without having to take more pills?

Yes w/Allergy shots. The approach to treat allergies is three-fold: avoid the offending allergen if possible (easiest for indoor allergens like pet dander); medications (like antihistamines & nasal sprays); and immunotherapy (shots) that decrease your body's sensitivity to airborne allergy provokers. When the first 2 approaches don't work or medicines cause side effects, shots are the best way to improve allergies.
Yes. It also may be that you are not on the right medications. Consider trying alternative antihistamines pills, or nasal antihistamines or steriods.