Advice for burn on my hand?

yesterday i picked up something from next to the fire without realising that there was hot coal and embers on it. they fell down my sleeve and got pushed into my hand for a while before i could get them out, it caused a lot of pain. i ran my hand under cold water for 10 minutes, and it was stinging and burning all night. today, it has not blistered much but has instead left me with a large, pink sore that is very painful, it is throbbing and it hurts to touch. there are black patches that wont wash off where it looks like the skin has been burnt. i didn't want to put cream on it as i heard this could seal the heat in? any advice on what i could do to relief the pain.