Using resonant inelastic x-ray scattering (RIXS) at the Cu $L$-absorption edge, we have observed intense, dispersive spin excitations in highly overdoped Tl$_2$Ba$_2$CuO$_{6+δ}$ (superconducting $T_c =6$ K), a compound whose normal-state charge transport and thermodynamic properties have been studied extensively and shown to exhibit canonical Fermi-liquid behavior. Complementary RIXS experiments on slightly overdoped Tl$_2$Ba$_2$CuO$_{6+δ}$ ($T_c =89$ K) and on Y$_{1-x}$Ca$_{x}$Ba$_2$Cu$_3$O$_{6+δ}$ compounds spanning a wide range of doping levels indicate that these excitations exhibit energies and energy-integrated spectral weights closely similar to those of antiferromagnetic magnons in undoped cuprates. The surprising coexistence of Fermi-liquid-like charge excitations and high-energy spin excitations reminiscent of antiferromagnetic insulators in highly overdoped compounds poses a challenge to current theoretical models of the cuprates.