= Solution
Use the same <method of multiple scales> and its <solvability condition in the method of multiple scales>. The <position-dependent damping> function is odd, so the lemma that <odd position-dependent damping has zero first-order amplitude drift> applies. In its <wave amplitude> average, $\psi\mapsto\psi+\pi$ changes $\sin(A\cos\psi)$ to its negative while preserving $\sin^2\psi$, so the average is zero. Hence $A'=0$; the <wave phase> average again gives $\theta'=0$.
Equivalently, expand the <position-dependent damping> in its uniformly convergent <power series> for bounded $A$. Each term $\cos^{2m+1}\psi\sin\psi$ has only even sine <Fourier harmonics>, so none resonates with the unit-frequency oscillator. With the initial <wave amplitude> and <wave phase> unchanged, \b[the leading uniform approximation is]
$$
\boxed{x(t;\epsilon)=\sqrt2\cos(t-\pi/4)+O(\epsilon)
=\cos t+\sin t+O(\epsilon),\qquad0\leq t\leq T_*/\epsilon.}
$$
There is no first-order slow drift. Effects at the next order accumulate only an $O(\epsilon)$ correction on this time scale.
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