Even though I have not had any family problems, I do exactly the same thing. I'm not sure why, but I think it's because I am very close to my family. If i ever see anything on television where somebody has left/died or a great expression of love, I immediately relate it to how I would feel in that situation and get all upset. Plus I am quite an emotional person anyway.
Maybe this could be something to do with your family problems. You haven;t specified what they were, but if they affected you deeply (and since family members are often the people with which you have the deepest and closest relationship), it could easily have triggered something.
I would not worry about it. Just because you are more affected by these things than you were before does not mean you are strange, it just means you are human.
And on the plus side, it's nice to find a guy who is able to connect with his emotions.