Could M-Theory explain dark matter as well as dark energy? Is it possible that M-Theory provides a solution to the mystery of dark matter and dark energy?  
The idea of dark matter and dark energy is that there is some inexplicable source affecting our universe.  
M-Theory suggests that there are other universes as well as our own.  If there were some degree of interaction between the universes, this could explain both the unaccounted for acceleration of our universe's expansion in which dark energy is suggested as well as the unidentified mass of our universe which is termed dark matter.  
If the other universes are orthogonal to some or all of our universe's dimensions, then both effects seem very possible.  Could the other universes' mass have some affect within our universe causing the gravitational effects on visible matter, radiation, and the large-scale structure of the universe? Could those universes also be causing the accelerated expansion of our universe by literally pulling it apart through similar gravitational effects.  
All of the effects and oddities seem like they would fall out of M-Theory easily.  The extra dimensions of M-Theory seem well-suited for explaining both effects.  Some dimensions could be effected by some alternate universes causing anomalies such as have been detected and relegated to dark matter.  While, maybe, all of the alternate universes could act as a gravitational pull on the very fabric of the universe through the more pedestrian three dimensions, as is suggested for dark energy.
 A: I want to start by clarifying what M-theory is, in relation to string theory, just so the context of my answer will be understood. These days, "string theory" encompasses five ten-dimensional superstring theories, the 11-dimensional M-theory of membranes, 26-dimensional bosonic string theory, "supercritical strings" in more than 10 dimensions, and other topics too. But the first six theories are the core of string theory as physics, and they have a lot of interrelations. In certain limits, the strings of one theory become equivalent to the strings of another theory. For a while, it was thought that M-theory would provide a unified description more fundamental than the five superstring theories, but for now, it's just another corner of the web of interrelations. 
The next thing to clarify is that to apply string theory as a description of the world, you need to be even more specific than picking one of the six theories. You might choose to work with "heterotic E8xE8 strings", but then you'll still need to choose which ten-dimensional space-time those strings inhabit. Generally that's a matter of choosing a six-dimensional "Calabi-Yau manifold" as the shape of the six spatial dimensions that we don't see. Your model of the world might be E8xE8 string theory on M^4 x CY6, where M^4 is four-dimensional space-time, and CY6 is the specific Calabi-Yau space. 
So having clarified a little how string theory is used as a physical theory... String theory models of physics, including those from its M-theory corner, are definitely meant to include dark matter and dark energy. But it doesn't usually work the way you suggest. In the vast majority of such models, dark matter is just another type of matter in this universe, and dark energy is vacuum energy, from quantum fluctuations in empty space. 
The closest I can come to your prescription is as follows. There is a branch of M-theory called heterotic M-theory, in which the 11-dimensional space-time is bounded by two 10-dimensional spaces. It's like a sandwich, the two 10-dimensional spaces are the slices of bread, and the eleventh dimension is the thickness of the sandwich. These two spaces on the boundary are like two literally parallel universes, each with its own internal matter and forces. But they can interact gravitationally, because gravity can cross the eleventh dimension. 
So in principle, one could have a model where the stars and galaxies of the visible universe exist in one of those boundary spaces, and the "dark matter" is matter in the other space, which because of shared gravity, has gathered around the places over there, corresponding to the galaxies over here. Accounting for dark energy as a gravitational interaction between the two spaces is harder, but it does seem to be a standard hope in the ekpyrotic model of bouncing universes, that the accelerated expansion which is attributed to dark energy, can be obtained in such a way. 
