We calculate the scalar self-force experienced by a scalar point-charge orbiting a Kerr black hole along $r\ensuremath{\theta}$-resonant geodesics. We use the self-force to calculate the averaged rate of change of the charge's orbital energy $⟨\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{E}⟩$, angular momentum $⟨{\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{L}}_{z}⟩$, and Carter constant $⟨\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{Q}⟩$, which together capture the leading-order adiabatic, secular evolution of the point-charge. Away from resonances, only the dissipative (time antisymmetric) components of the self-force contribute to $⟨\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{E}⟩$, $⟨{\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{L}}_{z}⟩$, and $⟨\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{Q}⟩$. We demonstrate, using a new numerical code, that during $r\ensuremath{\theta}$ resonances conservative (time symmetric) scalar perturbations also contribute to $⟨\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{Q}⟩$ and, thus, help drive the adiabatic evolution of the orbit. Furthermore, we observe that the relative impact of these conservative contributions to $⟨\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{Q}⟩$ is particularly strong for eccentric $2\ensuremath{\mathbin:}3$ resonances. These results provide the first conclusive numerical evidence that conservative scalar perturbations of Kerr spacetime are nonintegrable during $r\ensuremath{\theta}$ resonances.