I also don't talk about my stuttering to anyone too, my family is not supportive of my stuttering and I'm generally uncomfortable with the topic of it. Because yeah, it is embarrassing. However, these people are literally giving you a hard time at work and it's obviously taking a mental and emotional toll on you. You have every right to be able to go to work and not have to hear snickering behind your back constantly. You fear the shame attached to stuttering but honestly it's far more shameful to mock stuttering like your coworkers do, don't let them shame you anymore. Please consider taking a stand, write an email atleast to someone. You don't have to make it a big complaint if you'd rather avoid conflict, but its good to give someone a heads up to what's going on and how it's affecting you