If your insurance covers it (or you have the money), I would set up a consultation with an experienced Speech Language Pathologist (make they sure know about stuttering in all forms). If you can find a board certified specialist in stuttering, all the better. I do not know about your incident, but it certainly doesn't sound like developmental stuttering, which is the common type kids acquire. You could check over in r/slp, since I am not an speech language pathologist myself, but I wouldn't take your neurologists advice to just go and get meds (unless the meds are for other mental illnesses). Stuttering isn't caused by stress, although i'm not an expert neurogenic or psychogenic stuttering. I'd go see an SLP over a psychiatrist as they dont really know anything about stuttering. I have to go right now but the one thing I have learned after a lifetime of stuttering is, do not try to hide or avoid it at all costs (either through substituting words or not saying what you want). All this does is reinforce fear and the idea that "stuttering is bad" and once you identify with that internally, it can be a very tough road. Keep your spirits up and just keep talking, stuttering or not!