As photons bounce around and finally find their way into our eyes, are they continually relinquishing the information of the previous thing they "bounced off of"? Is this the reason why we receive a coherent, continuous image and not a chronologically chaotic image of the past meshed with the present?
It depends on what the photon is bouncing off of. For most materials, it's true that a reflected photon carries no information about what it interacted with prior to the reflection, but that's not the case for all materials.
To be precise, when a photon - actually an electromagnetic wave - is reflected, what really happens is that it gets absorbed by the reflecting material, and then a new EM wave is emitted to "replace" it. (Actually, it's more complicated than that; see this other question for example, but the re-emission explanation will suffice for now.) There are two properties of interest that could apply to the re-emitted wave:
In fact, for both these effects, the possibilities listed are only extremes of a continuum. A material can be partially reflective, for example, meaning that part of the absorbed EM wave is re-emitted in a specific direction and part of it is re-emitted randomly in any direction, as in window glass. In a case like this, you could say that the reflected light waves carry partial information about what they had previously been emitted from.