If you're having difficulty sharing posts as Public, it's probably due to some recent changes Facebook silently rolled out with regard to Application Privacy. By default, now, applications are only allowed to share with "Friends of Friends" and groups that are more restrictive.
If you would like to enable Public posting via dlvr.it, you will need to edit the application privacy settings inside Facebook for dlvr.it.
Log into Facebook
From the Account menu at the upper right, select "Account Settings"
Select "Applications" from the menu on the left
Locate dlvr.it in your list of applications and click on it
Change the "App Privacy" dropdown to "Everyone"
Changing this setting does not affect which setting dlvr.it uses for your Routes - it just allows dlvr.it to use the Public setting if you select it.