Sounds very uncomfortable!
In general, second degree burns should be evaluated by a medical professional.
Allergy to adhesives from bandages as well as topical antibiotics (such as neomycin or bacitracin) are common causes of irritation at the site of an injury.
For patients with concerns for irritation from adhesives, a non-stick gauze pad (e.g. Telfa) held in place with either paper tape or a self-adhering bandage (e.g., Coban or VetFlex) can be helpful.
Keeping the area moist with plain petrolatum (e.g. plain Vaseline) will speed the healing and prevent a dry sensation.
Hope this helps.