I think that these songs you're talking about help when you're down because they are talking to you emotionally. They are saying what you feel and making you feel less alone. It is proof other people struggle with the same problems.

Alternately, when you're not down they still talk to you about things you still understand because you've been there. So, it's like reminding you of where you've been and that's why you cry. It triggers the emotional response of sadness.

I think it's fully normal for that to happen. I have songs that I avoid at all costs. No matter where I am or how I am feeling, those songs will turn me into a blubbering mess and trigger me.