What is the name of your state?
It is important to note that, in the US, trademarks gain their rights through the use of the trademarks in commerce.
In the US, the first to use a mark is the presumed owner of the mark. This is not like other countries, where registration of the trademark shows ownership. In other words, federal registration with the USPTO is not a requirement.
For a trademark search, therefore, you can start with a TESS search but you must expand your search to include state registrations of marks and unregistered uses of marks.
I am not sure what you mean by a "design code for an ambigram." Are you planning to register a logo?