My takeaway is that you're pursuing a long-term relationship with someone who's not ready to commit due to deep-seated mental issues. Are you emotionally equipped and willing to take this arrangement for what it is rather than what you want it to be? Everything that you've mentioned till now seems to be indicative that you aren't. Keep in mind that there's no quick fix, and the situation isn't going to change overnight (if it ever does), so you can either adjust your expectations or leave while you still are on good terms.
She informed me of a guy that he manager at her job is trying to hook her up with and she told me she wasn't interested but because of the no official " label" of her and I she tells the dude she isn't interested in dating him. But the guy is persistent and keeps showing up to her work and texting her and because we aren't really official but she basically treats my like a significant other Im in a position where I don't really have any say over the whole thing.
Your ladyfriend's acquaintance refuses to leave her alone even though she's manifestly turned him down (saying you're not interested should be enough for someone to back off, mind you, no need to make clear you already "belong to" another guy). This is not a call for you to step in like a knight armed with jealousy. She should reach out to HR or get the authorities involved.