I agree with Paul in that it sounds like you're safe in this situation and should not encounter any problems.
As for the number they gave you, that should be your confirmation number from the cruise line and with that you can go onto the cruise line's website and verify your reservation and the information.
The agency should have provided you a copy of the confirmation the cruise line provided to them. If you did not get this, give them a call and ask them to send it to you. It's sent to them via an electronic document, so it's easy for them to forward a copy to your email address. Some agencies will only provide the confirmation from the cruise line, while other agencies will provide you two items; the confirmation from the cruise line as well as an invoice/confirmation from the agency.
Either way, you should always demand an confirmation from the cruise line and you should always verify it on the cruise line's website to insure all the information is correct. And if you notice any irregularities or errors, contact your agent immediately so they can correct them. You definitely do not want to wait until after the final payment due date. And if the agent wants to charge you for any changes, find another agent! (The only exception to this is if the cruise line is charging a fee for changes, but unless it's something like the Early Saver Fare offered by Carnival, almost always there should be no fee involved.)