Is the shape of an orbit unique for a given energy? Consider a body at a given distance from a star. If we project it perpendicular to the radius vector at the correct velocity, the body can undergo circular motion about the star. However, if we project it at an angle to the radius vector with the same velocity, I would expect it to undergo an elliptical orbit; however, will this orbit devolve into the circular orbit, i.e. for a given energy, is the orbit of the body unique?
 A: No.
For any given energy, there is a continuum of keplerian orbits that range from a fully circular orbit to more highly elliptical motions, all the way up to the limiting case of an almost-linear motion (extremely elliptical orbit) that whips around the focus infinitely sharply. These correspond to storing more or less energy in radial vs angular motion, i.e. to the angular momentum of the orbit.
There is also, of course, a three-dimensional degeneracy in the orientation of the orbit (usually given by three orbital elements such as the inclination, ascending node, and argument of periapsis) but dynamically speaking that is less interesting than the 'shape' degeneracy that comes from the variability in angular momentum.
And, for clarity, in a two-body problem, keplerian elliptical orbits are completely stable, and none of them "devolves" into any of the others. If there is a third body present then this changes (often quite dramatically) but then the space of possibilities becomes impossibly large to describe here.
A: Gravity, a central force proportional to $r^{-2}$, is conservative. So, assuming there are no other ways to lose energy (neglecting forms of friction, other bodies, etc.,) the orbit will never devolve. Diving straight into energy degeneracy, the total energy of a two body orbit is,
$$E=-G\frac{m_1+m_2}{2a}$$
where $a$ is the semi-major axis of orbit. This applies for all elliptical (bound) orbits. Thus, any orbit sharing the same $a$ will be of the same overall energy. 
A: 
will this [elliptical] orbit devolve into the circular orbit

In the simple case of two point masses, no.  The elliptical orbit is stable.  For extended objects (like the earth and satellites), tidal forces can exchange angular momentum and circularize the orbit.

for a given energy, is the orbit of the body unique?

No.  If you hold the energy constant, there are a range of orbits with different amounts of angular momentum and eccentricity.  The maximum angular momentum case (and minimal eccentricity) is the circular orbit.  
A: You kind of answered your own question. If we place an object a displacement $\textbf{r}$ away from a star, and give it an initial velocity of $\textbf{v}$ such that the angle between $\textbf{r}$ and $\textbf{v}$ is $\theta$, then $\theta$ parametrizes a continua of potential orbits (each value of $\theta$ yields of different orbit), each of which has the same energy (the total energy depends only on the magnitudes of $\textbf{r}$ and $\textbf{v}$, and not the angle between them).
So a simple answer to your question is no.
