A strong annulus theorem for 3-manifolds

I think this might follow from JSJ theory. Assume that $M$ is irreducible with incompressible boundary. Then any essential annulus is homotopic into an $I$-bundle region or a Seifert-fibered region of the JSJ decomposition. In the Seifert case, the region meets the boundary in annuli, in which case the boundaries of the immersed annulus will be a multiple of the core of the annuli, and I think one can cut and paste to get a simple closed essential core of each annulus, which then cobound an embedded essential annulus. Otherwise, if the annulis lies in a product region, then your immersed annulus goes between two curves on the surface. So this amounts to asking: for a given homotopy class of curve on a surface, can one always cut and paste a subset of the curve to give an embedded curve in the same isotopy class? This seems plausible, e.g., if the curve is not filling (one should be able to cut and paste to get a boundary component of the subsurface that it fills). In any case, it seems that at least this gives a reformulation of the problem (although I'm not sure what happens for twisted $I$-bundles).

I think that Theorem 15.1.5 in Scott's unpublished book on three-manifolds (email him for a copy) is very close to what you want.