We show that the SDE $dX_t = σ(X_{t-}) \, dL_t$, $X_0 \sim μ$ driven by a one-dimensional symnmetric $α$-stable Lévy process $(L_t)_{t \geq 0}$, $α\in (0,2]$, has a unique weak solution for any continuous function $σ: \mathbb{R} \to (0,\infty)$ which grows at most linearly. Our approach relies on random time changes of Feller processes. We study under which assumptions the random-time change of a Feller process is a conservative $C_b$-Feller process and prove the existence of a class of Feller processes with decomposable symbols. In particular, we establish new existence results for Feller processes with unbounded coefficients. As a by-product, we obtain a sufficient condition in terms of the symbol of a Feller process $(X_t)_{t \geq 0}$ for the perpetual integral $\int_{(0,\infty)} f(X_{s}) \, ds$ to be infinite almost surely.