Although this is a rather old question, I will try to answer it since the question implies some misunderstanding about plasma and tokamaks.
Let's start with plasma. In order to achieve fusion, two nuclei must be really close together - on the order of the nuclei radius itself. Only then the strong force will start to act, fusing the two nuclei into a new one. The problem is, however, that nuclei are positively charged and you need a lot of energy to overcome to repulsing Coulomb-force.
Energy means temperature and for the fusion reaction currently favored in a tokamak reactor, these are approximately 150 Mio Kelvin. Gas becomes completely ionized at these temperatures and we call that system no longer gas but plasma instead.
The main difference to a gas are the long-range interactions between the particles - in an ideal gas, particles only interact via (short-range) collisions. In a plasma, the particles interact via the (long-range) Coulomb-forces. This results for example in a collective bahaviour of the plasma particles and in the ability to screen test-charges you put into the plasma.
When you would now apply an external electric field, it would be somewhat shielded by the plasma particles. Hence, you can not use electric fields in a tokamak to efficiently confine the plasma. You use magnetic fields, since the particles' motion is affected by magnetic fields (Lorentz-force).
About the movement of the plasma: yes, the plasma moves but it is somewhat more complicated than you guessed in your question. It is true that you have a strong electric current flowing in the tokamak. And, as you explained, this also means that you have some ions flowing in one directions and electrons flowing in the other direction. But these are just a few particles. Due to the collective behaviour described above and small-scale instabilities (turbulent fluctuations), you have wave-like structures moving in the tokamak. They can rotate in one or the other direction. In addition, it is possible to inject high energetic particles in one or the other direction to locally drive currents (neutral beam injection). I guess what I am trying to say is that it is more complicated than it seems :)