I can spend an eternity with my girlfriend and not be bored of her, but saying goodbye is the problem. I could have just spent the whole day and the next morning with her and then after i go home I'm hit with a wave of anxiety, i can even feel it in my chest. Is this normal?
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An alternative could be to talk to her about it.
Anyways, I think that you're just very attached and there's really nothing too bad with that. Spend as much time with her as possible and make that time count. To me, it sounds like you're terribly in love ^^ Just hang out more and make sure you don't get too pushy and you'll be just fine, trust me!