All models that are fine-tuned are private to the account that requested the fine-tune.
Only the account that requested fine-tuned model has access to it and controls access to it, and yes you get a whole new fine_tuned_model like ada:ft-your-org:custom-model-name-2022-02-15-04-21-04
I would definitely recommend reading through docs on fine-tuning and embeddings before you proceed.
On a first look, it appears like embeddings could do the job for this case, but depends on the details.
Do you have developers on your team?