Behavior of the electric field of an incoherent light wave

I think that the thermal light is a good example. It is emitted by any object of temperature above absolute zero. Thermal light is a chaotic state, not described by a quantum state but by a diagonal density matrix whose elements describe the probability of different wavelengths, different polarizations, and in general no polarization is preferred, i.e. any polarization is equally probable.

However, I have a comment about your question, "Would the electric field oscillate with ever changing amplitudes and directions?". No, it's not true that there is a general electric field, of a certain type of polarization, but it changes direction very rapidly. You won't find such a common organization, you will find chaos.