Your formulation is misleading.
"Light is the way to exchange momentum between charge carriers".
First atoms are neutral, while light affect can only charge particles.
The atom is a composite of a positively charged point-like nucleous, and a negatively charged cloud-like charge density. This complex dance duo, can store energy in between them. Those are the bound states of the electron, but that's not the matter of electricity but chemistry. In electricity, this dance duo can store energy in its surrounding by deforming its electron cloud to become an electrostatic dipole.
Transaction is the wrong picture to have when we are dealing with electricity. The continuous picture is a lot better.
The momentum of an atom is a continuous quantity. At every moment in time it can be exchanged locally in continuous amounts. Both the positive nucleous and electron cloud are taking from the field and giving locally to the field.
Photon is kind of a confusing term because you have to distinguish between the virtual photon which mediates the coulomb interaction in a continuous way, and the real one which can go on its way, or be absorbed/emitted by atoms provided that the electron cloud can deform in such a way to account for energy conservation during the collision.
To see the distinction take the previous example of two protons H+ going towards each other then away. The trajectory to have in mind is they are following a perfect curves trajectories, and not a sequence of straight lines occasionally changing direction when the photon transaction happen. Those typical QM like trajectory you see in cloud chambers need the energy to become bounded in some discrete way. For example electron fly straight, real photon hit and is absorbed and atom change direction to conserve momentum and the electron jump to a higher orbital to conserve the energy (the energy is bounded to the atom for some time), the electron keep flying straight, then it emits a new photon and change direction. These kind of trajectories happen when the energy can only bounded in discrete quantities, but that's a matter about QM, and not electricity.
Finally clarifying what that the light field is carrying in : momentum, and making clear that the light field doesn't carry electric charge.