I do not have an "Add New" option in my boards, only "Add Task". How can I solve for this so I can apply templates to my projects?

Another way to add a task template is Convert task to template:

You can have task templates in project templates and thus make task templates available to new projects easily.

For existing projects with task templates, just re-home the resulting task to your destination project. That can be automated too. I explain that here:

Thanks,

Larry