If the E-mails are going out when you aren't on your computer, how is it your computer's problem? Sounds like your old provider has a problem.
My computer is rarely, if ever, shut down. It is just in rest mode. A habit I have now stopped. I assume that is why it can happen.
The "sent" emails don't show up in my sent box. I would have no way of knowing about them if it wasn't for the fact they were failing.