Yes, you're over-thinking it, send the text NOW and ask them for coffee asap. If you keep putting it off, the worst case scenario is that this other girl ends up asking them first and then you'll be stuck with no bridesmaids!! If you've already been engaged for 3 months they won't think you're trying to "get in front of" the other girl, you're already ahead in the "queue" anyway!
I know exactly how you're feeling, I ended up over-thinking and putting off my bridesmaid proposals until it was ridiculous. One of them I only ended up finally asking because my mum was literally kicking me under the table to ask her. Before that I'd kept convincing myself that it wasn't the right moment, or other things were more pressing, or agonising over what various people would think if I asked this person at this moment... So I have been there, and trust me, I can tell you that you are indeed overthinking it.
If you don't end up managing to arrange a coffee, pop them a nice card in the post or something. People love that sort of stuff, and once it's in the letterbox you can't take back out!