Superman23, I don't presume to know your exact situation, and I've only had to move once in my life (though Pennsylvania to Texas is a big one, I'd say), but I think we have the same issue, and someone telling you that friends come and go, don't worry about it, doesn't really help. But god I wish it did.
Kourin is right, though: fear of abandonment is fear of being rejected in a different form. Part of the problem is that a lot of people were raised with this picture of idealized friendship: Best Friends Forever! If you haven't been friends with someone for years and years, it doesn't have the same perceived weight. And if you don't have a Best Friend, then it's because no one has deemed you worthy enough of it, and have rejected you for this honor above all honors.
It seems like you understand that thinking like that isn't quite right. Acting to negate thoughts thoughts and the actions that follow from them is the hard part, but at least you've got a start, right?
I belong to an avoidant social network called AVPD Net. PM me if you'd like an invite--I don't bite!