Do you have Outlook installed on the machine you are trying to do this from? Can you send successfully through Outlook (Mapi)? Have you ascertained that the settings for the Direct method are identical to the settings in Outlook?

Having your IT person explain to someone who has no idea what they're talking about is sometimes a way to discover a problem.

We can send them through Outlook but it is cumbersome: We get a warning message twice, and have to click on "allow" twice, and then the emails get stuck in my outbox. We then have to go in and forward all of the emails (you can't just hit "send" again - it isn't an option), one by one. My understanding is that if we send them directly it is much easier.

It is. But if Outlook can "talk" to your SMTP prvider, then Adagio should be able to as well. Perhaps there's a firewall issue? Some anti-virus software prevents "unknown" applications from sending e-mails. You might want to look there too.

Did the Test Email worked?

Assuming that you are sure that "all" the outlook (including port) info is the same, have you tried:
(1) Logging on to the account via a browser using your webmail access. If for some reason you "think" you are certain that you know the password (which is the only setting that you can't sight verify) that Outlook is being successful with but you are actually using something different - then attempting to logon through your webmail access will show this up.
(2) Just changing the password to something new and updating Outlook to use the new password - see that Outlook's test is successful and then try the same in Adagio. Again if the problem is a bad username / password combination this will show that up.
(3) Turn off all Firewall Security and re-try. If no luck, turn off all FW and AV security and re-try.
(4) Try temporarily using a different account that is based on a different mail server - your IT guy can help with this

Still no luck, print and mail in an envelope that says "you've got email"

Pete

Check this registry setting change to make that annoying message go away. Worked for us.

http://sogeeky.blogspot.com/2007/04/how-to-disable-outlook-security-warning.html