My girlfriend broke my heart a year ago

So a year ago my girlfriend broke up with me and completely broke my heart. She had good reasons to and I have no ill feelings towards her for making the choice she did. My job and behaviour had driven her too this and trust me it wasn't easy for her to make that decision! We had been going out for 2 years and were living together at the time. I handled it badly at first. I did everything wrong by texting and calling her all the time and practically begging her to reconsider. Finally I realised the error of my ways and got help getting myself back on track. I now have a new career and most importantly of all an new outlook on life. I have tried dating, even had a short relationship with someone, but no one has even come close to my ex. I now know, more than ever that she really was the one for me and I completely took her for granted. We have, until recently stayed in touch. Even spent time together having dinner a couple of time which was really great. But she told me a few months ago that she had started seeing someone. At first it crushed me but I quickly realised that that would be incredibly hypocritical of me as that was exactly what I had been doing and that I didn't blame her. Still it changed nothing for me. I love her, I miss her and I'm prepared to do anything it takes to bring our lives together again. Oh and before you say, yes I know there's a big chance she won't want me and she's over me. Trust me I have tried talking myself out of this hundreds of times but I still come back to the same answer, I want her back. My question is how? As cowardly as it sounds I'm frightened of contacting her as I don't want to come across as needy and desperate as I once did. Still I want to talk to her so much, face to face if possible as I feel that would be much better and phone, text or letter. I know I'm not the only one who has been in this situation so I'm reaching out for some advice. Anything is welcome.