We numerically establish the controllable conversion between Laguerre-Gaussian and Hermite-Gaussian solitons in nonlinear media featuring parabolic and cross-phase potential wells. The parabolic potential maintains the stability of Laguerre-Gaussian and Hermite-Gaussian beams, while the actual conversion between the two modes is facilitated by the cross-phase potential, which induces an additional phase shift. By flexibly engineering the range of the cross-phase potential well, various higher-mode solitons can be generated at desired distances. Beams carrying orbital angular momentum can also be efficiently controlled by this method. In addition, other types of beams, such as sine complex-various-function Gaussian and hypergeometric-Gaussian vortex beams, can be periodically transformed and manipulated in a similar manner. Our approach allows the intricate internal relationships between different modes of beams to be conveniently revealed.