Well done, OP, for having the strength to walk away from what sounds like a pretty loveless marriage. I hope you do get your shot at happiness, as you certainly deserve it, and if you've found a nice, emotionally available, respectful man - that's great.
However. I do think in your understandable excitement, you have rushed things where your DD is concerned. Some of what is happening sounds like normal 13yo hormones, but mostly it sounds like a very unhappy and insecure young girl trying to get her mum to listen to her fears.
Take a step back, OP. Put aside the defensiveness and the resentment and re-read, as objectively as you can, the posts from PP's who were in similar positions as children, and the damage that has done to their parental relationships over time. They're not saying these things to be cruel, they're trying to give you a warning. You're at a crossroads here with your relationship with your daughter and it could go either way depending on your next move.
You know what I'd do if I were you? I'd speak to your new man very frankly. I'd say something like "I really like you, I really want to carry on seeing you and I'm excited about where this might go. But at the same time I think I've rushed things along too much with my children and for their sake, and for the sake of having the best shot of us all being happy together down the line, I need to slow it right down. Date night once a week out of the house, a couple of sessions on Skype per week at weekday lunchtimes. Are you okay with that?" (Note no hint of blaming your daughter's behaviour for YOUR decision here.) If he is the lovely wannabe-family man you think and need him to be, he'll be absolutely fine and you'll know you've got a keeper. If he goes off on one, becomes petulant or tries to put pressure on you, then your daughter's perception will be proven correct.
I'm not pretending it will be easy, OP, because when I suggest slowing it right down, I mean for months at least, possibly even years - as long as it takes for your DD to trust that you do put her first and to accept that Mr New Man is not going to come between you both - which, to be fair, it sounds as though he has to an extent up to now. And you need to do this with a happy heart and the desire to help your daughter have the happiness in childhood she is entitled to - no resentment, no pressure, no trying to hurry things back to "happy families" before she's ready.
If there needed to be, there could be other men for you in the future. But your daughter only has one mum. Don't be the kind of mum who puts being with a man, any man, above their child's happiness and peace of mind. I'm sure you're not that type of woman. It's important your daughter can be sure of it too.