I've talked some friends into using <3 instead of :( to express sympathy, for exactly this reason.

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.

There is "Hugging Face", but it looks just weird (IMHO) and I don't know many people using it.

https://emojipedia.org/hugging-face/

That looks more like it's communicating heartfelt happiness or something. I've never understood it as a hugging face ...

seems such an emoji is actually part of the newest standard as I just saw someone mention on Twitter: https://emojipedia.org/people-hugging/

If it's your friend, doesn't it make sense that she would feel sad when your are not feeling well? Empathy is being able to feel someone else pain.

Understanding someone else's pain makes sense, feeling it doesn't.

The last thing you need when not feeling well is causing more suffering.

It's inevitable, there is no need to feel guilty about it.

It's kind of an extreme form of vegetarianism if somebody doesn't show suffering because he fears that others would suffer from compassion. A friend in need is a friend indeed. Whoever suffers with you doesn't mind the suffering.

The definition of empathy is feeling another's pain

What else is compassion? If somebody just acknowledges your emotions are they really close and a friend?