I think your situation is very common, and very normal! You are NOT overreacting!! As an early childhood teacher, I always like to reassure parents that most of the time, the separation is harder on the parents than it is on the child!!
Even for children who have experience leaving their parents, there is always the potential for separation anxiety. This is a very normal and appropriate stage of development for 3 year olds and you have to remember a few very important things-
1. He will sense your anxiety, and he may cry the first couple times you drop him off. You will probably cry, too. But it only gets better as he becomes more comfortable and develops trusting relationships with the teachers. You have to trust that he will adjust and be fine. (He will!)
2. If he does have separation anxiety, the best way to overcome it is to make it fast and consistent. I always tell my families that I want them to come, help their child get settled, etc. But when it is time to say goodbye, just say goodbye and leave. If you prolong the goodbye, it prolongs the tears.
3. Talk to the teacher and see if she will call or if it is okay for you to call and check on him the first few times. Chances are he will be fine, and the teacher will hopefully be more than happy to have you call. (If you hang around, there is a chance he will see you and it will be all over from there, so I suggest leaving and calling after about 15 minutes.)
4. DO NOT "sneak" out when he isn't looking. You should ALWAYS say goodbye, even if it causes a meltdown. If you think the goodbye meltdown is bad, what do you think he feels like when he REALIZES you are gone and has no idea when you are coming back??? Just a tip from an experienced preschool teacher. :)
My daughter was 14 months old when I had to take her to child care 2 days a week. It was seriously the worst form of parental torture (for me), and I am an early childhood teacher!!! She cried for the first 2 days until it was time to take a nap, but each day got better and better. I'd say at 14 months, it took her about 2 months to fully adjust, but then she'd walk right in and go do her thing. Now she is almost 3, still goes twice a week, and can't stop talking about her friends and teachers.
So it IS true that they overcome the separation and sometimes forget to even say goodbye to you, but I don't want you to think that it will be this way from the first day, especially because it might not be.
Best of luck to you and your son- he will do great and you will too. And you will LOVE all the things he learns and does by being away from you a couple times a week, making friends, and learning to love school!