Evolution on changing fitness landscapes (seascapes) is an important problem in evolutionary biology. We consider the Moran model of finite population evolution with selection in a randomly changing, dynamic environment. In the model, each individual has one of the two alleles, wild type or mutant. We calculate the fixation probability by making a proper ansatz for the logarithm of fixation probabilities. This method has been used previously to solve the analogous problem for the Wright-Fisher model. The fixation probability is related to the solution of a third-order algebraic equation (for the logarithm of fixation probability). We consider the strong interference of landscape fluctuations, sampling, and selection when the fixation process cannot be described by the mean fitness. Such an effect appears if the mutant allele has a higher fitness in one landscape and a lower fitness in another, compared with the wild type, and the product of effective population size and fitness is large. We provide a generalization of the Kimura formula for the fixation probability that applies to these cases. When the mutant allele has a fitness (dis-)advantage in both landscapes, the fixation probability is described by the mean fitness.