Good evening and welcome. I am very sorry to hear about these false allegations.
To address your question, employers actually have a legal obligation to investigate allegations of sexual harassment. So, they would be correct in looking into the matter and inquiring with you about it. That said, the law does not require an employer to believe all allegations of sexual harassment or to take adverse employment action against an employee simply because they were accused.
The other overriding legal principle here is that employment in NH is "at will" absent an express agreement to the contrary. At will employment can be terminated for any reason not amounting to discrimination on the basis of a legally protected trait (race, religion, gender, etc.) or retaliation for engaging in certain forms of legally protected conduct (filing a wage claim, taking FMLA leave, etc.). It doesn't matter whether the basis for termination is fair, reasonable or even TRUE.
Putting this all together, employers are required to investigate allegations of sexual harassment but they are not required to believe those allegations if there is no evidence to support them. However, if an employer does believe the allegations, they are free to terminate an employee on the basis of them, regardless of whether their conclusion is fair or accurate.
Thus, your only real power here is the power of persuasion. This means cooperating with your employer's investigation and offering any facts or opinions that may help them seem that these allegations are the fabrication of a disgruntled former employee. You cannot, however, force your employer to not investigate, nor would you typically have any legal claim against them if they decide to terminate your employment based on their investigation.
I hope that you find this information helpful. Please do not hesitate to let me know if you have any questions or concerns regarding the above and I will be more than happy to assist you further.
If you do not require any further assistance, please be so kind as to provide a positive rating of my service so that I may receive credit for assisting you. Very best wishes moving forward.