You are hanging on to her, you are being clingy so to say. Imagine if you had a person that is a bit unstable and has problems in their life, now that person so happens to get to know you and you are friendly to them. All of the sudden that person won't leave you alone, you sense that they might want something more than a platonic relationship (Where it is completely fine to hang out for 6 months and then just kind of not hang out that much, it is ok in platonic relationships). Now because you are an empathetic person you realize that they are betting way too much of their happiness onto you, and you are unsure of this whole relationship to begin with. So you start to distance yourself because you don't want to hurt them anymore than you need to.
You have already demonstrated that you don't quite have your shit together so to speak. But it is ok, you don't need to have your shit together. But you absolutely need to be able to handle your life on your own, and not bet all of your happiness on another person. That will create immense pressure on the other party, and since they cannot respond with the same kind of attachment they will distance themselves.
You need to get your shit together, you need to be able to send a couple of texts and then not send anything for weeks if that person doesn't respond. You need to be ok with them fading away completely. It will be counter-intuitive at first, but this will actually cause people to relax around you. They will know that even if THEY screw up somehow you won't go off and kill yourself. You are a person that can handle shit. Since you aren't there yet this is what you need to build on, and this is what all young men have to go through. It is a part of the rite of passage to becoming a man.
Being clingy will make you more alone than ever. Being relaxed and forgiving will give you friends. The saying "Happiness comes from inside" is about this, it's about developing yourself and your life so that you can function alone. Then it will be so much easier for people to hang around you and not feel the burden of you clinging on to them and you hoping that they will make you happy. They can sense this a mile away, especially women.
Don't worry, you can do this! And always remember to give people space when they signal to you that they need it. (Refering to the "I get the feeling she didnt actually want to spend much time around me".) Here you lay back and let them come to you, or contact them again in a couple of weeks and see where you are at. If it doesn't go back from there you need to start letting that relationship go. It could also turn to something more infrequent for a longer while, which you are ok with if you so want of course.