Your kids' dramatic reactions make me think they have a lot of fear around this. I would want to give them an opportunity (in counselling or privately) to voice exactly what those fears are. Really hear them out, and then address their fears one by one. You don't plan to divorce. You don't plan to abandon your family. You're not going to make them accept your girlfriend as another mother, or spend time with her if they don't enjoy her company. They won't have to witness any displays of affection that feel icky or inappropriate to them. Their peers aren't going to find out and tease them about this. You're not going to burn in hell. See if you can pinpoint what it is that upsets them so much. If this is not even a new relationship, you can point out that nothing is changing.
I agree that dumping your girlfriend for their sake is not a wise choice. They should know that you don't treat loved ones that way, but you can try to make your time with her as discreet as possible as they get used to it.
I wonder how my kids would take it...
Married to a monogamous man 15 yrs, mother of 2, dating C 3 yrs, and in a romantic friendship with L more than 20 yrs