When you quantize Maxwell’s equations to get QED, the equations still hold. All that changes is the meaning of the potentials and fields; they are now quantum-mechanical operators. The field of a charge still extends arbitrarily far from the charge.
In fact, the quantum electromagnetic field exists even when there are no charges and no photons. For example, it has “quantum fluctuations” in the vacuum state. And a Big Bang does not require charges to produce photons; expanding spacetime is sufficient to produce them. (By the way, the quantum electron-positron field similarly exists when there are no electrons or positrons. Fields are more fundamental than particles, at least in mymany physicists’ ontology.)
Your conception of photons as being the field is incorrect. They are the quanta of the field. These quanta interact with matter as point particles, but the quantum field that they are quanta of exists everywhere, just like classical field in classical electromagnetism.
Finally, each charge does not have its own field. There is only only one quantum EM field pervading the universe, and all photons in the universe are quanta of this one field.