The answer has to be 'not necessarily', because, in general, as you go along, you're free to adjust the solar sail angle and thus the trajectory. In addition, the trajectory need not lie in a single plane since the sail can produce out of plane forces.
I posted an analysis in the comments yesterday for a weak solar sail and a shallow spiral orbit and concluded that the orbit was a logarithmic spiral. But I think the analysis extends no matter how steep the spiral.
A first point to note is that both the gravitational force and the solar wind force decay as $1/r^2$ and thus keep the same ratio irrespective of radial distance. If we assume the sail is set at a fixed angle with respect to the radial direction, then not only is the ratio of the two forces constant but their respective directions are also independent of radius. This already suggests a constant angle or logarithmic spiral. A logarithmic spiral is 'self-similar' and looks the same at any scale of radius, $r$, i.e. its features scale with $r$.
The only remaining question is whether the forces (i.e. acceleration, $a$) and velocity, $v$, change together in a commensurate manner as the radius changes. The quantity $v^2/a$ has units of distance and therefore must also scale as $r$. Since $a$ scales as $1/r^2$, it follows that $v$ scales as $1/\sqrt(r)$. The radius of curvature of the curve is given by the velocity-squared divided by the perpendicular component of acceleration. It follows that the radius of curvature is proportional to r, further confirming the logarithmic spiral.
Having concluded that exponential spiral orbits exist, we should note that they are defined by a single parameter. In that sense they are similar to circular orbits. They are a special case in that presumably you must start with exactly the right velocity at the right position in order to continue on a desired spiral. Arbitary starting conditions will not generally yield a logarithmic spiral orbit any more than they might yield an exactly circular orbit.