I think some people have the wrong view about piano, and instruments that are similar, when the issue of "hand independence" is discussed. Cognitive science these days seems to indicate that humans never truly "multitask" in their conscious actions; they simply switch between individual tasks very quickly. While I can imagine a human with an unusual neurological feature (such as a severed corpus callosum) performing true multitasking (such as typing out a history paper while having an unrelated conversation), the overwhelming portion of humans are neurologically incapable of such an action. In fact, it is the hallmark of an unhealthy human that parts of their brain do not fire in a coordinated concert of individual (i.e., singular, sequential) neurological explosions, but instead fire independently of each other, of their own accord. This is actually what you see in the brain of a patient who is having a seizure. All the areas of the malfunctiong brain ignore each other and just fire at their own will; in other words, the areas of their brain act independently of each other.
Therefore, you do not want true "hand independence"; instead, you want hand coordination. Put another way, instrument playing is a conscious action, controlled by our executive function, and we only have one area of the brain that controls the executive function. Thus, homo sapiens's conscious control is, for better or worse, unitary, and we cannot do two independent tasks at once.
The same is true for the piano.
So...what you need to do is to try to not beat yourself up too much, and simply accept the fact that when you do hands-together play you really are learning a different skill. You cannot simply focus on the movement of one arm, wrist, hand and finger in isolation from any other. If you can't play it with hands together, go back, define every individual action you must take, treating both hands as a single mental unit, put them in sequence, practice them over and over again until you can do them in sequence, in time, perfectly, and then speed it up. Do not try to skip this simple process because you want your hands to "be independent", because you cannot neurologically do such a thing.