We study a real-valued Lévy-type process $X$, which is locally $α$-stable in the sense that its jump kernel is a combination of a `principal' (state dependent) $α$-stable part with a `residual' lower order part. We show that under mild conditions on the local characteristics of a process (the jump kernel and the velocity field) the process is uniquely defined, is Markov, and has the strong Feller property. We approximate $X$ in law by a non-linear regression $\widetilde X^x_{t}=\mathfrak{f}_t(x)+t^{1/α}U^{x}_t$ with a deterministic regressor term $\mathfrak{f}_t(x)$ and $α$-stable innovation term $U^{x}_t$, and provide error estimates for such an approximation. A case study is performed, revealing different types of assumptions which lead to various choices of regressor/innovation terms and various types of the estimates. The assumptions are quite general, cover the super-critical case $α<1$, and allow non-symmetry of the Lévy kernel and unboundedness of the drift coefficient.