Do the stars of the Hulse-Taylor binary follow geodesics? The Wikipedia definition of a geodesic is,

the worldline of a particle free from all external, non-gravitational forces

Now for a binary system like Hulse-Taylor I have heard it described as a free-falling oscillating mass quadrupole and so the worldlines are perturbed from the geodesic. How accurate is this? It seems to me that if we have two stars that are just under the influence of gravity then their motion must follow geodesics?
 A: The orbits are following geodesics, which are pretty much ellipses, just the geodesics can have minor perturbations that are still consistent with General Relativity. That is, the full orbits, incorporating all effects, would have some perturbations from elliptical, which makes it a geodesic of the total spacetime metric.
Still, the slight perturbations to the orbits have not been measured. What has been measured is the effect of the rotation on the X Ray frequencies observed.  
The oscillations you are referring to are probably the quasi period oscillations (QPOs) of the X Ray frequencies that have been observed in many compact stars such as neutron stars and black holes. In the last few years it's been possible to determine the effects well enough to check that it is consistent with General Relativity. For tight binaries the effect is larger, but it's is generally due to the so called Lens Thirring effect. 
See the article at http://sci.esa.int/xmm-newton/58072-gravitational-vortex-provides-new-way-to-study-matter-close-to-a-black-hole/
The QPOs are quasi periodic small changes in the frequency of the X rays. In the case reported in the article it was a black hole, and the frequency flickered, with variations in the frequency. The article that determined the effect was published in May 2016. See it at  https://academic.oup.com/mnras/article-abstract/461/2/1967/2608396/A-quasi-periodic-modulation-of-the-iron-line?redirectedFrom=fulltext
The Lens Thirring effect is due to frame dragging in a rotating spacetime. Near the neutron star or black hole the frames of reference rotate. That is part of what is known as one effect of rotation, and it is also try for instance in the Ker metric for a rotating spherical symmetric body. See the description of the year general effect at https://en.m.wikipedia.org/wiki/Lense–Thirring_precession
