Yes, whether you are on the Free, Pro or Expert plan, Design Wizard will save your designs.
Click on the save icon on the top right hand side of your workspace and your design will be saved in your ‘My designs’ folder.
You can also save a duplicate version of your design by clicking on the drop down menu beside the save button. Once you click on this arrow you will be given an option to 'save as new design'. Click on the 'save as new design' button and your design will be saved in your 'My designs' folder.