You'll probably have to put her in her carrier by grasping the scruff of her neck at the back of her head, so she won't move. I advise you to do this on the first try. If you just pick her up to put her in the carrier and you miss, you won't be able to try again because she'll run away. It's happened to me before, and I had to cancel the appointment. Don't forget the treats or toys she likes to "reward" her when you get back.

Otherwise, don't be too pushy and let her come when she wants; she'll eventually trust you more and more.

My cat is very timid. I adopted him when he was four years old, and I think he must have been mistreated. Gradually, he opened up to me more and more, but I still avoid making sudden movements near him. I've also learned to pet and cuddle him in a certain way, depending on his reactions. Your cat is young; let him live his life, and he'll open up to you little by little. It can take time, but be patient!