A modified version of the replica method recently proposed by:the authors [J. Phys. A 32, 585 (1999)] is applied to a certain class of models describing the motion of a particle in a random potential with superposed parabolic confinement, Different types of randomness are considered. Previous treatments of analogous models were based on replica symmetry breaking (RSB). The aim of the paper is to demonstrate that the new method provides reliable results for the average free energy and related quantities without taking into account RSB. The comparison with computer simulations shows satisfactory agreement. [S1063-165X(99)00810-7].