You can make changes to your classic Signup Form without creating a new form or changing the code for the form on your site.
To do so, go to your Audience tab within your Emma account and scroll down to signup forms. Then click the tab for Classic forms (you may need to scroll down again after doing this). Click the down arrow to the right of your form and select "Edit this signup."
Once you do so you can make any changes you like to your signup form (including changes what group the contacts go into).
Make sure to click Save after making any changes. Once saved, the changes will take affect immediately.