This is what I have imagined electricity to be like, where the blue balls represent the free valence electrons in a conductor atom, and the electrons flow towards the positive electrode as the electrons flow from lower to higher potential, propelled by the electric forces acting on them. The electrons enter the cell through the positive electrode, get pushed to the negative electrode by some chemical reactions and then get pushed out as and why I stated before.
Here, the electrodes of the cell are still in contact with the conductor, but they do not have a path to flow through smoothly as the circuit wire is broken. However, the EMF of the cell still exists, so shouldn't the electrons try to flow away from the negative terminal, and the electrons on the part of the wire flow into the positive terminal and then pass out from the other end of the cell and do the same, eventually accumulating at the broken end of the wire connected to the negative terminal, since they have don't have a path to go through?
Also, if this is true, does it mean that if I touch that broken wire without being connected to the ground or conductor of any sort, a transient current will still flow through me?