There would be some context, wouldn't there?! She wouldn't just say "Here's a phone number - call it if your name is Sophie and you live in Wigan."
Hi Sophie - I don't know if the email below is relevant to you. Of course just ignore if not.
Best,
Management
Hello - I exchanged details with a lady called Sophie in Costa Coffee on xx street in Wigan last Friday. We intended to stay in contact, but I've now lost the piece of paper with her phone number. Unfortunately, I didn't get her surname, but she mentioned that she worked for xx organisation. I would be grateful if you could pass my number onto anyone at your organisation called Sophie who might've been in the area that day. Much appreciated. OP 07- ----
There. I don't think that's going to frighten anyone.
And if the organisation doesn't want to send it on, they don't. But OP has given it her best shot. No harm done.
I think what is being totally missed here is the wider context - that OP is a foster carer in need of support. She found shared experience with another foster carer - they had a lot of challenges in common, and chatted for 2 hours. A connection was made - finding that can be huge, and sometimes you do need to do whatever you can to follow these things up. If this new friendship can help OP be a better foster carer - it's worth the risk of some random organisation thinking she's a bit bonkers.
So many exceptional things I've achieved in my career and my life have come from following a gut instinct and being a bit "dog with a bone" - continuing to push at a door and taking a shot in the dark on the off chance. The fact something is unlikely to work, doesn't mean you shouldn't try, if it's not going to do anyone any harm.
[The call in Costa was a very bad move - I agree. But that's a side bar, and the OP clearly regrets it. I don't think we need to write her off because of it.]