"My therapist did say that she felt like she would be really affected by me if I killed myself--could she be protecting herself and that's why she won't be my therapist? Do therapists think that way?"

There is your answer. I am a retired therapist and yes we do think like that. Her not wanting to see you is more about her than it is about you. She may have had a client suicide recently and doesn't feel she isready to talk to someone with SI. Another possibility, she may have things going on in her RL and she may feel she doesn't have the emotional energy.
She is taking care of herself. She is modeling good boundaries. Again this is more about her than you.

Hoping your new therapist works out well,

Sabra