hi bpgf, and

it sounds like you are both on a different page regarding the terms of the relationship:
So I'm suspicious. Especially with me in a place of wantin to be in a relationship. I want someone true & loyal.
wanted to try again. But things had to be on her terms. She wants a relationship... .but she's still hurt & wants to move slow. Well that didn't happen.
... .
it made me feel like we are together.
three months in, in a revived relationship that had an eight month split is a fragile time (how long were you together the first time around?). the honeymoon phase can be good, and youll spend a lot of that time remembering and reminding each other about what you loved. it sounds like youd been spending a lot of that time together (she met your father and reconnected with your kids). i get the sense that you, understandably, took all of this as a sign of increased commitment and investment in the relationship on both ends. three months in is also a stage where pressures, and being on a different page can threaten a relationship. given she has indicated she needs space, i think thats where it stands.
Before you can make anything better, you have to stop making it worse.
your girlfriend is communicating that she needs to build trust in order to establish a relationship. you are suspicious and acting on that suspicion - not a one time thing, but each time that youre looking for a sign of commitment. this isnt building trust. its not providing incentive for her to commit. if she needs space, it is pushing her away.
it will help inform your next steps if you can fill us in on this:
So tell us what happened in detail and what you said. Also, did you apologize very clearly? What did you say?