If the files are appearing in your Sent Files view, then I'm very confident that the emails were sent. It's possible that they got trapped by the recipient's spam/junk filter.
This problem seems to be on the increase -- it happened to me over the weekend when purchasing theater tickets with e-mail delivery. When I didn't get the e-mail within an hour, I called the box office and they suggested I look in my spam folder (gmail). There they were.
One thing you mentioned is puzzling, though. You're saying that the files are showing as "sent" on Adobe Send Now? Did you mean Adobe Send?
It's Adobe Send. That's good advice, but the one I sent myself as a test, with the same email as the customer, is definitely not in the spam folder - so I will have to call Adobe today. Thanks for trying to help.