We propose a new, to the best of our knowledge, and very general finite power beam solution to the paraxial wave equation (PWE) in Cartesian coordinates by introducing an exponential differential operator on the existing PWE solution and term it as the "finite-energy generalized Olver beam." Applying the analytical expressions for the field distributions, we study the evolution of intensity, centroid, and variance of these beams during free-space propagation. Our findings demonstrate that these new beams exhibit a diffraction-resistant profile along a curved trajectory when specific beam conditions are met. Using numerical methods, we further demonstrate the ability to adjust the self-accelerating degree, sidelobe profile, and stability of the central mainlobe by manipulating the transforming parameters. This research presents a versatile approach to controlling beam properties and holds promise for advancing applications in various fields.