Axion dark matter (DM) may convert to radio-frequency electromagnetic radiation in the strong magnetic fields around neutron stars. The radio signature of such a process would be an ultra-narrow spectral peak at a frequency determined by the mass of the axion particle. We analyze data we collected from the Robert C. Byrd Green Bank Telescope in the L-band and the Effelsberg 100-m Telescope in the L-Band and S-band from a number of sources expected to produce bright signals of axion-photon conversion, including the Galactic Center of the Milky Way and the nearby isolated neutron stars RX J0720.4-3125 and RX J0806.4-4123. We find no evidence for axion DM and are able to set some of the strongest constraints to-date on the existence of axion DM in the highly-motivated mass range between ~5-11 $\mu$eV.