Do modern dispersion-corrected DFT methods give more accurate molecular geometries?
I am afraid I do not know of a paper specifically discussing the effects of DFT dispersion corrections on intramolecular properties. However, it should be noted that for short-range descriptors (such as bond lengths and bending angles) the dispersion corrections will not influence those as they correspond to distances below the typical distance at which the corrections take place. The various schemes differ in how they work, but typically the corrections (in Grimme's schemes) are damped below a distance of ~6 Å. It is also my experience that such local structural characteristics are not affected by dispersion corrections.

However, you are right that dispersion corrections will influence intramolecular structure at larger scale, and in particular for “soft” or “flexible” long molecules with many conformations. There, it is worth noting that the so-called “dispersion” corrections actually correspond to the lack of accounting in local DFT functionals of middle-to-long range correlation effects. Thus, it is actually expected that the corrections — or nonlocal functionals including dispersion terms — actually perform better for those too.

Unless, like Grimme's 2006 two-body dispersion correction scheme (often called “D2”), the dispersion corrections happen to be often too strong and make intramolecular distance somewhat too small (instead of much too large)!



Edit: Now that I come to think of it, one extreme example of intramolecular system that has been studied in detail w.r.t. dispersion corrections is that of helicenes. It was shown (for example there and there) that dispersion corrections improve the accuracy of computational structures and energies compared to experimental (or high-level computational) data.