If you've deployed both the Analytics and Audience Manager tools via DTM, you should see network requests for both those solutions on your web property. You can view these requests via the network tab in the browser's developer tools, or by using your favorite debugging plugin. To verify that the integration is working, you should first check to make sure you are seeing both network requests. After you've verified the network request, check to see if the data is available to you in Audience Manager. There are a few documentation links below that may help.
My knowledge of Audience Manager is limited to the integration implementation, so I apologize if my answer isn't detailed enough. Let me know if there are additional questions and I'll check with some of our Audience Manager experts.
The best way to integrate Adobe Analytics and Adobe Audience Manager is via server-side forwarding. Have a look at this documentation. To deploy server-side forwarding using DTM, you'd need to do the following:
Contact customer care with the tracking servers being used by this implementation and ask them to enable server-side forwarding on the tracking servers
Ensure that your site has deployed Marketing Cloud ID Service (MCID)
Remove the Audience Manager Tool from DTM
Change you Analytics tool in DTM to be custom
Deploy the Audience Managment module inside the Analytics library
Test and deploy
Please let me know if you have additional questions.