Incorrect spelling, explanation: when we add -ing suffix to words that end with e, this e is skipped, so sue becomes suing, not sueing. Just like other words of this case, for example, dancing or smiling, suing is the correct spelling. Please keep in mind, though, that there are exceptions to this rule.

Correct spelling, explanation: this form is the right one because, in case of adding -ing ending to the words with e, we omit the last e. Therefore a gerund form of sue is suing, not sueing. Suing is the correct spelling.

Definition of suing:
verb, gerund form of sue, to start a law process against someone
Today Jamie is finally suing her employer for sexual assault.
Have you heard that they are suing two other companies?