Hi Ryan,
I think you are on the right track with the nanoflow. What you want is in the text box you can use the on change event. With that event you can trigger a microflow or nanoflow, then in your microflow or nanoflow, use the change activity to change the attribute of your search field.
You can also add additional logic to check if the attribute is empty before setting the new value if the search field, that way you can prevent the case that the user types a value in the search field and then types something into the text box.
Hope this helps!