= Solution
Introduce $T=\epsilon t$ and write $x_0=A(T)\cos[t+\theta(T)]$. Averaging
$$
\frac d{dt}\frac{x^2+\dot x^2}{2}=-\epsilon\dot x^4
$$
over one fast period gives $AA_T=-3A^4/8$ and $\theta_T=0$. The initial data therefore give the <method of multiple scales> result
$$
\boxed{x(t)\sim\frac{\cos t}{\sqrt{1+3\epsilon t/4}}}
$$
through $t=O(\epsilon^{-1})$.
For the replacement damping, $\sin(\dot x)\dot x=\dot x^2-\dot x^4/6+\cdots$ is even in $\dot x$. Every term has zero resonant projection onto the fundamental over a complete orbit, so the $O(\epsilon)$ slow amplitude and phase equations vanish. Thus
$$
\boxed{x(t)=\cos t+O(\epsilon)}
$$
through $t=O(\epsilon^{-1})$: there is no first-order secular damping, although bounded mean and higher-harmonic corrections occur.
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