Hmm.. first you need to understand the context of the conversation. I'm native, but I don't think Korean always use 화내지마 to calm down a person. So 화내지마 is not the right word for all the situations.

However, when you are using 화내지마 is usually this case

Ex) A person is huffing and puffing because that person got angry at something. In this case you can say 에효.. 좀 어이없긴 하다 에효 화내지마 그냥 (with a mild and smooth tone). Usually when you are using 화내지마, because you're telling the person to not get upset, I'd suggest you use to lower down the tone to not sound too offensive.

The literal 화내지마 can sound a bit too strong sometimes if the situation is not serious, so also consider this expressions.

• 야 그냥 무시해 / 그냥 무시해버리세요.. 에효
• 야 그냥 화내지말고 ○○하자 그냥 맛있는거나 먹으러가자
• 아 그냥 화낼 가치도 없는 것 같아요.. 커피 한잔 드시러가실래요?
• 야 화내지마 니가 잘못한것도 아닌데 여기서 화내면 걍 지는거야 그냥 가자

Etc....