Disclaimer: I am not a doctor, do not take this as medical advice.
Basic first aid for 2nd/3rd degree burns (source, source) is to:
- Cool the burn with clean, cool water. Avoid ice and don't cool off extensive burns (>20% of body surface) because of the risk of hypothermia.
- Remove clothing/jewelry from affected areas.
- Keep the skin clean and dry.
- Cover with a clean, dry dressing, preferably sterile and non-stick.
- Avoid popping blisters unless they are infected.
You'll notice the word "clean" in there a lot, and that's because the biggest danger with a severe burn is infection. Burned skin loses its defense against bacteria, so keeping the wound clean will be your primary concern. In an apocalyptic scenario, without proper antibiotics, infections can easily turn deadly through septic shock.
The use of antibiotic ointments on severe burns is controversial. Some handbooks recommend Silvadene cream, but others discourage putting any ointment on the wound.
Definitive treatment in normal circumstances would include surgical debridement and skin grafts, but neither is likely to be practical in an apocalyptic situation. Without proper supplies and training, you'd probably do more harm than good. Larval therapy is an option, but eeew!
The Survival Medicine Handbook has a whole chapter on natural burn remedies, from vinegar to honey and cotton-ash paste. It's too much to summarize here, but the book is a great resource for apocalyptic fiction writers.
A suture won't take the place of a skin graft, and in fact can compound the risk of infection.