You probably
are making her feel sad when you tell her. That's how sympathy and empathy work, it's feeling sorrow or pity for someone's misfortune (sympathy), or feeling a pain similar to theirs (empathy). The whole point of telling your friends and loved ones is so you feel
slightly better at their
slight expense. Do it with enough friends, and you might feel significantly better, but individually they are all only slightly worse. That's part of the implicit job description of being a friend, being there to help in that way.
If I knew a friend purposefully didn't share that with me for my sake, I would be a little offended if I viewed them as a good friend, as obviously they didn't view me as a good enough friend to let me help in that way.
I wouldn't let that hug fool you either. In real life if you tell someone your problem and it makes them want to hug you, that hug is probably preceded by a frown in reality as well.