Is there a connection between anxiety and stuttering, stammering?

Yes and no. Anxiety can produce stuttering - & stuttering certainly produces anxiety. But speech dysfluencies can &often do exist independently of underlying anxiety. To me, stuttering is one of the more difficult conditions to treat. And it can be heartbreaking. A comprehensive approach is essential. Too often, meds are not considered, but they should be, if only for the secondary anxiety involved.
A difference. Anxiety does create a hesitation in speech that resembles suffering, however, stuttering and stammering, a speech impediment starts in childhood and may well be related to the mother-child relationship, an anxious mother transmits her anxiety as the child is learning to speak. Psychiatric treatment reveals that hostile feeling can cause stuttering.