In short, legally you do not need to file either — but it’s a good idea to do so. United States copyright law for visual art is considered fully and legally copyrighted upon creation.
Filing with the government grants you additional rights in case of infringement (such as being able to have damages awarded to you). In other words, if you don’t file, you can sue an infringer for money.
The artwork I create for you would fall under copyright laws, not trademark laws. Please see my in-depth article on the differences between copyrights and trademarks when it comes to cartoon logos and characters for detailed information.
Please consult with an actual lawyer for legal advice, and one that knows the intellectual property laws.