It seems to me that you can’t send email because your server can’t resolve MX records of recipients.
Without struggling too much with your intentions, needs or scenario, You might be able to route your emails if you configure a Smart Host in your exchange server.
This way your server will forward every single internal generated email to that smart host. This smart host will then “search the internet” (via DNS) for whom is responsible to receive emails for destination domain.
You can configure a smart host in:
Exchange System Manager \ Administrative Groups \ First Adm. Group \ Servers \ YOUR SERVER NAME \ Routing Groups \ Connectors \<b> Internet Mail SMTP Connector (SERVER)</b>
Right click it and choose Properties.
In General tab choose “Forward all mail through this connector to the following smart hosts” and configure the smart host you want to use.
I think your best bet for this is to ask your Internet provider which server you should use and put it here.
I hope this makes sense to you. If you need further assistance, please post back.