You should keep the area constantly moist and the best thing you can do BY FAR is to keep the wound constantly moist with colloidal silver.  Though mainstream medicine tends to disavow and villify silver because of the huge threat it represents to their billions in annual profits from patentend antibiotics, major burn centers use silver burn ointments and silver bandages extensively - often with the result that even third degree burns heal without scarring.

Silver will prevent any infection and will greatlly aid the healing process, likely because as Dr. Robert Becker reported, silver has the ability to induce fibrocytes to dedifferentiate into stem cells.

NOTE: the colloidal silver works even better in combination with aloe vera - but you should get a fresh aloe vera plant and cut out the inner leaf fillets instead of relying on commercial products.

Other recommendations would be manuka honey, lavender essential oil and potato slices (Cut a potato in half and put it on the burn area.  The starch neutralizes the burn, pain and future scarring. Also works great on sunburns.)  Though I would save the potato remedy until after you have used one or more of the other items for initial healing, given the more serious nature of the burns as you described them.

One thing that helps speed virtually all healing is the supplement Bromelain, which is an extract of the pineapple plant that you take internally.

Do not use cotton directly on burns as it adheres to it.  Do not apply Ink, Mercurochrome, greases or baking soda over burnt area, and do not try to peel it loose.

After the wound has healed, taking plenty of digestive enzymes on an empty stomach two to three times a day can help eliminate and lessen any residual scarring.