We examine the Standard Model field configurations near cosmic strings in a particular class of models. This class is defined by the condition that the generator of the flux in the string, Ts, commutes with the Standard Model Lie algebra. We find that if the Standard Model Higgs carries a charge Fh/2 under Ts, cosmic string solutions have Z-flux Fz = [n - FhN/(Fφ)]4π cosθw/(g) , where n is any integer and4πN/(qFφ) is the flux of the gauge field associated with Ts. Only the configuration with the smallest value of n -FhN/(Fφ) is stable, however. We argue that the instabilities found at higher Fz are just associated with paths in configuration space reducing n -FhN/(Fφ) by one unit. This contradicts recent claims that the instabilities in such models represent the spontaneous generation of current along the string. We also show that if the stable cosmic electroweak string has a winding in the electroweak Higgs field, then there are Standard Model fermion zero modes on the string. However, recent work suggests that these introduce new instabilities on the string.