Every couple has to negotiate what is comfortable for both partners for both private and public expressions of love and affection. I’m delighted that you two have found each other and can be affectionate in a way that is pleasing to the pair of you when you are alone.
There are two issues to consider regarding affection in front of the kids: First, it’s generally a good idea for dating single parents to go slow until you are reasonably certain you are going to have a future together. It’s very confusing to kids when they are asked to repeatedly open themselves to the idea of mom or dad having a new partner, only to have the latest someone “special” disappear.
In addition, you have your girlfriend’s sensitivities to consider. She admits she was traumatized by her own early experience. It seems she has generalized her younger self’s shock to include all forms of physical affection between two adults.
I do think she would benefit from talking that out with an experienced counselor – for two reasons: She is depriving herself of all forms of affection except when she is absolutely sure kids aren’t around. She is therefore missing out on the little moments of reassurance and love that provide emotional sustenance to any relationship.
Just as important is that she is depriving the two girls of a role model for what healthy, respectful affection between two adults looks like. Kids, being kids, are taking “notes” all the time. They are watching the two of you to see what they are supposed to do when they have romantic relationships of their own. Something to seriously consider is whether you want the girls to grow up unable to hold hands, hug, and give a kiss goodnight without feeling they are doing something wrong.
I suggest you talk this over with her. There’s no “right” answer. But I do think it’s important to make a clear and conscious decision rather than let a childhood experience dictate the present.
I wish you well.