Could a "living planet" alter its own trajectory only by changing its shape?

If you allow for non-Newtonian gravity (i.e., general relativity), then an extended body can "swim" through spacetime using cyclic deformations. See the 2003 paper "Swimming in Spacetime: Motion by Cyclic Changes in Body Shape" (Science, vol. 299, p. 1865) and the 2007 paper "Extended-body effects in cosmological spacetimes" (Classical and Quantum Gravity, vol. 24, p. 5161).

Even in Newtonian gravity, it appears to be possible. The second paper above cited "Reactionless orbital propulsion using tether deployment" (Acta Astronautica, v. 26, p. 307 (1992).) Unfortunately, the paper is paywalled and I can't access the full text; but here's the abstract:

A satellite in orbit can propel itself by retracting and deploying a length of the tether, with an expenditure of energy but with no use of on-board reaction mass, as shown by Landis and Hrach in a previous paper. The orbit can be raised, lowered, or the orbital position changed, by reaction against the gravitational gradient. Energy is added to or removed from the orbit by pumping the tether length in the same way as pumping a swing. Examples of tether propulsion in orbit without use of reaction mass are discussed, including: (1) using tether extension to reposition a satellite in orbit without fuel expenditure by extending a mass on the end of a tether; (2) using a tether for eccentricity pumping to add energy to the orbit for boosting and orbital transfer; and (3) length modulation of a spinning tether to transfer angular momentum between the orbit and tether spin, thus allowing changes in orbital angular momentum.

If anyone wants to look at the article and edit this answer accordingly with a more detailed summary, feel free. As pointed out by Jules in the comments, the "previous paper" mentioned in the abstract appears to be this one, which is freely available.

The idea of "swimming in spacetime" was also discussed on StackExchange here and here.

Conservation of angular momentum tells us that in an isolated system, total angular momentum remains constant in both magnitude and direction.

The key here is that the conserved quantity is the total angular momentum: spin+orbital angular momentum.

An example:

For a planet, angular momentum is distributed between the spin of the planet and its revolution in its orbit, and these are often exchanged by various mechanisms. The conservation of angular momentum in the Earth–Moon system results in the transfer of angular momentum from Earth to Moon, due to tidal torque the Moon exerts on the Earth. This in turn results in the slowing down of the rotation rate of Earth, at about 65.7 nanoseconds per day, and in gradual increase of the radius of Moon's orbit, at about 3.82 centimeters per year.

(source: Wikipedia)

Let's suppose that Solaris' sun does not rotate. If Solaris' spin axis' direction is $\vec n$, the total angular momentum will be

$$\vec L_\text{total} = \vec L_\text{spin} + \vec L_\text{orbital} = I \omega \ \vec n + M r^2 \Omega \ \vec k $$

Where $\omega$ is the spin angular velocity, $\Omega$ the orbital angular velocity and $r$ the distance between Solaris and its sun.

So if Solaris is able to change its moment of inertia $I$ by changing its mass distribution, we see that it is indeed possible for it to adjust its trajectory, because if $I$ changes then $\omega, \Omega$ and $r$ will have to change to conserve total angular momentum.

A different mechanism: On a long timescale, by increasing the surface area exposed to the sun (flattening the planet), the radiation pressure would increase, boosting to a higher orbit. Changing the albedo would be a more effective means to the same end but could allow assymetric force as well Either way it would be simpler in a tidally-locked planet. This has been proposed for deflecting asteroids. Extrapolating from figure 3 at that link, a perfectly reflective surface of the same scale as the asteroid/planet would take millenia for enough deflection to avoid an asteroid/coemt hitting Earth. There doesn't appear to be a limit to the timescale in the question, so assuming geological timescales this might be what you're looking for.