A few months ago I had a falling out with a friend I really adored and trusted- she really hurt me and never apologized or tried to reconcile. I miss having a good friend / mom friend and have felt so sad- but don't want to seem needy. I met someone I really clicked with. How do I not screw up?

What was the falling out? I kind of can't imagine that as an adult. Mostly it's the slow fade (though some of the fades have made me very sad as well).
after she was mean, and I told her what she said was hurtful, she got defensive, I just faded. But she hasn't tried to make it right. FWIW it was my therapist who told me to cut her off ... she was even more shocked by her behavior than I was.
I think just take your time with the friendship and be patient. Make plans but don't be all over this person. I get very turned off when I meet someone that I really like and then they try to be very close too soon. Real friendships take time. If you really click now, you will really click months from now. I know it is hard when you lose a friend but I do think when someone leaves your life like what you described, it is usually for the best. It probably hurts now, but you will feel better sooner than you think!