When I got laid off - I switched to a cheaper brand of sinus rinse packets and developed cracks just inside my nose on the outside corners. I went back to the NeilMed packets and even stopped rinsing for several days - but the painful cracks have not healed.
The NeilMed saline solution should not cause any burning if used according to our directions because it is pH balanced, meaning it matches the concentration of saline in your body. You may want to let the cracking heal before doing more sinus rinses. Consider using a product like NealMed NasoGel. It is a water soluble saline gel that has aloe vera and other ingredients that help moisturize dry nasal passages.