You can use the routing tools and directions template publicly. In this case, I think that the workflow is like this (I'm trying to find the DOC):
a. Right now, your org is configured to use the default routing service. Your browser is passing the credentials of your current ORG account to the routing server.
b. What needs to happen in this case is to use ArcGIS.com to create a proxy to automatically pass your organization credentials when using routing - but understand that this will consume credits.
c. To do that, you can add this as an item to your 'My Content' on ArcGIS.com: https://route.arcgis.com/arcgis/rest/services/World/Route/NAServer
d. When you press 'OK', you should be prompted for credentials. Provide your named user credentials, and choose the option to 'save credentials'. This will create the proxy.
e. Now when the item is created, in the bottom right of the items detail page, you should use a URL. It should read like the URL is for https://utility.arcgis.com/usrsvcs... - not https://route.arcgis.com. Use the 'copy' button to copy this URL.
f. Finally, in My Organization>Edit settings>utility services, paste the URL copied in step 'e' above in to the 'Directions and Routing' combo box.
I'm unsure which changes, if any, you'd need to adjust on the app side. It may just be to reconfigure the routing widget to use the URL from https://utility.arcgis.com, as that URL routes through the proxy you created when you saved the credentials with the item.