I sent 3 emails asking when can we start with the official procedures of preparing and signing the contract. I always received the same answer 'whenever you want', I would reply that I need to start it early so I can settle and finish the boring paperwork, then I receive nothing back.
Based on what you've written here, it sounds like one possible explanation for these event is simple miscommunication: the professor told you that she was willing to start the procedures any time you wanted, but you never replied with a specific date, so she just assumed that you weren't quite ready yet, and that you'd tell her when you were.
If you did, in fact, want to start immediately, you should have told her so; instead, you used vague expressions like "early", which the professor may have interpreted as "soon, but not yet".
That said, even if this is the real explanation, I wouldn't call it entirely your fault; the professor should have become concerned when you told her that you wanted to start the procedures early, and then never followed up with a "let's start now", and she should have written back to you sooner to explicitly ask you whether you still need it done early, and if so, to name a specific deadline. Still, it's quite possible and understandable that she may have been distracted by other, more pressing matters, especially if you weren't in frequent communication and if there was no particular deadline involved for her.
Regarding what you should do now, I agree with the other answers that the damage is basically done already. One option, if you want to take it, is to simply walk away and take one of the other offers. You may be leaving a burned bridge behind you if you do so — but that bridge is already aflame, and there's no guarantee that you can put it out at this point.
If you do want to try and mend your relationship with this professor, what you need to do is sit down and talk with her. Explain what you believe has happened, from your viewpoint, and ask her to explain how she sees the situation, and be willing to accept that those two viewpoints might look very different, and that the misunderstanding may not be entirely her fault.
Before you do that, it may be a good idea to make up your mind on which postdoc position you want to choose, and make that clear to her from the beginning. If you're going to take one of the other offers, make that clear from the start, and say you're sorry — don't give the impression that you're trying to make the professor bargain to make you stay. And conversely, if you do want to stay and work with her, also let her know that from the start. Either way, make it clear that this isn't a negotiation, but that you just want to figure out what went wrong and how you can avoid making the same mistakes again.
(That's not to say that you should irrevocably commit to a particular choice before the discussion; it is, after all, possible that some new information might come up during or after the discussion that could legitimately make you re-evaluate your choices. But do at least make a definite choice first, and make it clear that you're not expecting to change your mind unless something truly unexpected comes up. The point is that you're not sitting down to bargain, but to clear the air and figure out how to move on.)