Can I change the branding of the form to my personal branding? (not the domain, the design)

If you need to fully customize the form, you can also use its full source code and re-design everything.

If you can provide more details on what "personal branding" you were after, we will be able to provide you a more appropriate suggestion/solution. 

Also, just in case it was the branding logo in your form, you will be able to remove that when you upgrade to paid plans.

Refer to this thread as well for example.

https://www.jotform.com/answers/919642-Is-custom-branding-available-for-JotForm-customers-on-all-plans-If-not-which-plans-is-this-feature-available-on

Let us know if you need further assistance.