Sorry to hear about the trouble you are having. In order to transfer any domain, they have to confirm ownership by emailing the primary address listed for the domain. This is to ensure nobody takes someone else's domain.
You can definitely register a different domain in the meantime, but it will use your free domain credit (if your plan included one). Also, keep in mind that if your domain expires and is released, there is a chance that it will be picked up by someone else.
I looked on domainsbyproxy.com, and you can request a new password by clicking Login, then Forgot your login information?.
Then enter your domain name, and type the text displayed in the image below in the Enter security code: field.
It will then send an email to the non-private address they have on file for this domain.
I went ahead and performed these steps for you, so you will hopefully receive a password recovery email.
Domainsbyproxy also offers live 24 hour support.
If you have any further questions, feel free to post them below.
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