I’m impressed with how generous and mature you are; especially since you are only 13. Instead of being mad at dad, you are trying to figure things out. Good for you!
I suspect that your dad starting to date has stirred up your feelings of loss and pain about your mom’s death. This is perfectly normal. You miss your mom. I don’t think the negative feelings are personal about the girlfriend. My guess is that this would happen no matter who she is.
Your reactions also suggest that you are re-experencing the traumatic way your mother died. Sometimes little kids kind of protect themselves by numbing feelings that seem too overwhelming to feel. Then something happens that opens up those feelings again. As strange as it may seem, the fact that you are doing that is healthy as it suggests that you are now strong enough to handle it.
I think you need to do two things: First – have a quiet talk with your dad. You can tell him that you are happy that he is feeling ready to move on but that it’s hard for you. Let him know that you know that it probably would be hard no matter how nice the woman is. In fact, the nicer she is, the harder it might be. Then – tell him that you know you have some more work to do. Tell him what you told me and ask him to help you find a therapist who specializes in helping teens with trauma and grief.
With some support and good advice from a caring professional, you can learn how to both honor your mother’s memory and make room for an adult woman friend who loves your dad.
I wish you well.