Interesting story in the making!
According to https://isogg.org/wiki/Autosomal_DNA_statistics#mediaviewer/File:Shared_cM_Project_v2_updated.png
your 357-cM match is weaker than would be expected for a first cousin -- it's more consistent with a second cousin or a first cousin once removed.
Testing a first cousin would be a way to learn more about the possible relationship, but unless that particular cousin turns out to have a really close relationship (such as a half sibling) to this black-market baby, it's likely that you'll need to test more people than just one cousin to get to the answer. It would be a good idea for both you and the black-market baby to upload your data to Gedmatch, where you might discover some more distant cousins that you both relate to, and where you'll have access to the data you may need to map the DNA matches you find.
You say that all the aunts and uncles on your father's side are deceased, but it's possible that this person's parent is somewhere else in your family, so you will need to be mindful of the sensitivities of living people.