I solved this, by blundering around TSVN. Assuming you want your working folder to be Interface\AddOns\MyNewProject, this is how to fix the missing trunk, tags, and branches folders in your repository URL.
Note, this is for brand new projects, not for projects that have been converted from the old system.
Navigate to Interface\AddOns, but not inside the \AddOns\ folder. You do not want to see the list of installed AddOns, but the parent folder that contains your AddOns.
Right click and pull up TortoiseSVN's repo browser from the submenu.
Right click on AddOns, select SVN Checkout and make sure you have the URL correct from step 8.
Code and profit!
Unlike the old WA/CF, the subfolders for tags, branches, and trunk are not automatically created when you set up your project's Source. This could change in the future, but for now, hopefully this helps someone.