  Spin excitations are explored in the electron-doped spin-orbit Mott insulator
(Sr$_{1-x}$La$_{x}$)$_3$Ir$_2$O$_7$. As this bilayer square lattice system is
doped into the metallic regime, long-range antiferromagnetism vanishes, yet a
spectrum of gapped spin excitation remains. Excitation lifetimes are strongly
damped with increasing carrier concentration, and the energy integrated
spectral weight becomes nearly momentum independent as static spin order is
suppressed. Local magnetic moments, absent in the parent system, grow in
metallic samples and approach values consistent with one $J=\frac{1}{2}$
impurity per electron doped. Our combined data suggest that the magnetic
spectra of metallic (Sr$_{1-x}$La$_{x}$)$_3$Ir$_2$O$_7$ are best described by
excitations out of a disordered dimer state.
