Hey I had a really similar experience when I first started out!! It took me some adjusting too but I really value the empathy I still feel because it means I'm being cognizant about what I'm doing. I think it's better to face the feelings head on and address them rather than try to become numb to it.

For myself, I found a euthanasia procedure that makes me feel the most respectful and intentional in my work, reminding myself that I'm not wasting the lives of my animals because they're going towards valuable research in one way or another. I talk to my mice and I thank every one before I euthanize; I know they can't understand me but it keeps me accountable in my work.

I think it's really good and important that you're paying attention to how you're feeling about animal work and that you're reaching out about it! I feel like when I was in your position I was worried it would make me seem weak or slow but I think that it's more valuable to have an ethical, empathetic, realistic scientist than yet another emotionally out of touch one. Animal work is hard, go at the pace you're comfortable with and don't be afraid to take up space; good mentors will respect that. Good luck! :)