With a nonadiabatic dynamical method the polaron dynamics in organic ferromagnets with spin radicals is investigated under weak electric fields. The results reveal two novel phenomena different from those in normal polymers due to the existence of spin radicals. One is that the velocity of the polaron is asymmetric upon the reversal of the applied electric field, which is explained from the asymmetric polarity of the polaron charge density in different directions of the field, and hence its effect on the lattice distortion. The other is the 'intermittent rebound' of the polaron, where the polaron intermittently moves against the electric field force during a short interval behaving like a negative current. The details of lattice distortion and charge distribution of the polaron during the process have been revealed. We further found that there exist different critical fields for the above two phenomena. With an increase of the electric field, the 'intermittent rebound' of the polaron vanishes first and subsequently the asymmetric polaron velocity. This work demonstrates the unique properties of polaron transport in organic ferromagnets, and will be helpful in the future design of organic ferromagnetic devices.