What could be the reasons for (and remedies to clear up) dry, painful scabbing inside the nostrils?

Many. Dry nose and scabbing can be due to excessive picking at nose, allergies, bacterial infections, and others. Avoid picking the nose, use saline nasal sprays, humidification, a small amount of vaseline in the nose and consider the use of intranasal antibiotics. You may need to see your pcp or ENT for evaluation. Humidification, humidification, humidification is the answer. Lots of saline spray!
Superficial infxn. Most likely a superficial infection of the linings of your nose. Commonly caused by staph. You can try over the counter Bacitracin placed inside your nose 4 times a day with a clean finger in addition to this magical over the counter solution call ponaris oil. Use the ponaris twice a day as directed. You should see improvement within a few days. Do not use saline as this can irritate this problem.