Hello and hope you are doing well.
Your symptoms could be due to a sinus infection, a cold or a chronic allergy. Your symptom of nose bleeds may also indicate a bleeding disorder. The causes for bleeding disorders are hemophilia, von willebrand's disease, immune disorders, leukemia, aplastic anemia and scurvy. The bleeding can also be due to liver disorders, certain medications like blood thinners and non steroidal anti inflammatory agents. This can be detected by blood counts and bleeding and clotting time. Please discuss this with your doctor am sure he will provide further assistance.
Hope this helped and do keep us posted.
Sounds like the lining of the nasal passages are very dry and irritated.
If the nose bleeds are persistent then you should be consulting your physician.