Immediately after the burn, run it under cool-lukewarm water. Not cold, not hot. You may or may not get a blister. If you do, don’t pop it. Simply bandage it with nonstick wraps, the liquid in the blister helps your skin heal. If it pops treat it like an open wound. Aloe is good to use. It can sting like shit if the burn is too fresh, but does help after. Better yet is an aloe based burn gel. After initial healing, keep it moisturised as you said.

Not a doctor or medical person, but I am a very experienced fire performer and have tended to all sorts of burns.

Once you’ve cooled it and applied polysporin, you should put a bandage to keep from smearing off all the polysporin onto your clothes and keep it clean. If you don’t have a bandage that clean long sleeve will do, but it might get a greasy mark you’ll need to treat later for laundry: try rubbing in a dab of Dawn before laundering it.

Bear in mind, it may randomly start hurting and feeling burny again later, because the nerve endings hate burns. Cold water will help again. Fridge temp is fine, helps numb it without causing ice damage.

Add celery, onion and a potato.

Keep it under running water for ½ an hour, at the level that is the most comfortable.
Don’t make it too cold, but cold enough to ease the burning sensation.

If the pain stops before ½ an hour, you can stop.

Salve/creme/foam containing panthenol (in Europe it’s Bepanten, not sure about the US). This stuff does wonders helping your skin recover. Great thing against sunburns, will work with a usual burn as well

What that other person said about water.

On top of that, you’ll know the severity of the burn in a few hours. Most likely a light 1st degree burn (red itchy skin), at worst a light 2nd degree (blistering).

If it remains just red, treat it with some ointment (Bepanthen is my go-to). Be sure to keep it clean. Some minor blistering might occur, which is fine if it’s not bothersome. If blistering becomes severe, cover with wet bandages and see a doctor.

Do not scratch!

So, can we get an update as to the severity? Is it looking like it’s gonna be an amputation or a band-aid?