Katie, I'll definitely be thinking of you, and based on everything you have said I am very optimistic that you are totally fine and caught everything very, very early.
To answer your question about whether other people have felt the same way you did when they first found out: I have never had any suspicious moles or growths, but I do have a friend who did, and was told it could be melanoma. She definitely went into a panic, and actually ended up seeing a therapist and going on some basic anti-anxiety medication because the prospect of having a malignant melanoma was so scary to her. Like you, she has two young children, and she said that hearing that she might have something potentially deadly was terrifying to her. It turned out that she was totally fine, the growth was easily removed, and she has had no further issues. She does go for skin checks every six months, just to be extra safe, and I know that makes her feel a lot better. She is a completely normal, rational, and very intelligent person, and I think her reaction was completely understandable, as is yours. There is absolutely no reason to apologize for how you are feeling.