Hello, I'd like to help you with your problem.
It sounds like you are aware of what happened in your friendship and have tried to repair the damage. Because this was a new friendship, it may be that your friend thinks you are like this all the time, having no other reference to go on. Also, you don't have much history together so she may be considering that as well. That could be why she has backed off.
You have done the right thing in sending her a few cards apologizing for what happened. That was a good first step. You may also want to try telling her exactly what you said in your post, that you are scared that you have ruined the relationship for good. If you have not told her that, she may be open to hearing that you value the friendship that much.
Also try focusing on her for a while. Don't mention what happened but instead ask her about her life. For example, ask if you can see some of the pictures from her trip and see if she will tell you what it was like for her. Listen and do not bring yourself into the conversation at all. Focusing totally on her lets her see that you are interested in her and that you value her experiences as well.
Talk with your mutual friend and ask for help. Tell your mutual friend that you want to try to start over with your friendship and ask if she knows of anything you can do to make things better. See if she is willing to talk to your friend and let her know how badly you feel about what happened. If your friend values your mutual friend's opinion, that can work well in your favor.
If you try to repair this friendship and it does not work out, you can still take something positive from the experience. You learned that if something significant happens to you, that you feel you can turn to your friends for help. That is not a negative but a positive in that you have a good connection to those around you. You also can plan ahead. If you know that you need a lot of support when you go through a difficult time, then it might help to still go to your friends for support but also plan on seeking therapy or some other type of support so you have more that one source to help you through. This experience helped you learn more about yourself and that is always a positive.
I hope this has helped you,
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