Research shows that, depending on the group, ritalin can both ease and worsen stutter. Looks like for some people stuttering is indeed caused by dopamine dysregulation, while for others it's something else. It's not an amphetamine but can be considered adjacent. For me personally ritalin did wonders and I have the same reaction to caffeine as you. Also a mix of theanine, thyrosine and taurine works somewhat well if I don't have access to ritalin. I've also been prescribed atarax (hydroxyzine hydrochloride) for my stutter and it did help but made me very slow and I couldn't think properly when I was on it (enough for everyday functioning or reading lectures but not enough for coding or math). In the west it's used as an antihistamine, in Russia they focus more on anxiolytic/sedative properties. Perhaps some other anxiolytics could also work.