We present the results of a mid-infrared survey of 26 asteroids in comet-like orbits, including six Damocloids and six near-Earth asteroids (NEAs). We define a ``comet-like'' orbit as one having a Tisserand invariant TJ under 3 (but only including objects that are NEAs or otherwise unusual). Visible-wavelength data were also obtained, so geometric albedos (in the Cousins R band) and effective radii are presented for 25 objects (plus one more with 3 σ limits) as derived using the NEA Thermal Model. Nine of our objects were observed at two or more mid-infrared wavelengths, and in all cases the low-thermal inertia thermal model was found to be applicable, with various values of the beaming parameter. Our work more than quintuples the total number of observationally constrained albedos among TJ<3 asteroids to 32. Defining the ``comet-like'' albedos as those below 0.075, we find that 64%+/-5% of the sample has comet-like albedos. Objects in comet-like orbits with comet-like albedos are candidates for being dormant or extinct comets. Indeed, one of our targets, 2001 OG108, became active again a few months after our observations. We find a very strong correlation between the albedo distribution and TJ, with the percentage of dark TJ<3 asteroids being much greater than that of the TJ>3 NEAs. There are 10 NEAs among the 32 objects, and of those, 53%+/-9% have comet-like albedos. With the current crop of NEAs, this implies that about 4% of all known NEAs are extinct comets. A comparison of the histogram of TJ<3 asteroid albedos with that of active cometary nuclei shows that the former has a larger spread.