Differential expression analysis of sequence-count expression data involves performing a large number of hypothesis tests that compare the expression count data of each gene or transcript across two or more biological conditions. The assumptions of any specific hypothesis-testing method will probably not be valid for each of a very large number of genes. Thus, computational evaluation of assumptions should be incorporated into the analysis to select an appropriate hypothesis-testing method for each gene. Here, we generalize earlier work to introduce two novel procedures that use estimates of the empirical Bayesian probability (EBP) of overdispersion to select or combine results of a standard Poisson likelihood ratio test and a quasi-likelihood test for each gene. These EBP-based procedures simultaneously evaluate the Poisson-distribution assumption and account for multiple testing. With adequate power to detect overdispersion, the new procedures select the standard likelihood test for each gene with Poisson-distributed counts and the quasi-likelihood test for each gene with overdispersed counts. The new procedures outperformed previously published methods in many simulation studies. We also present a real-data analysis example and discuss how the framework used to develop the new procedures may be generalized to further enhance performance. An R code library that implements the methods is freely available at www.stjuderesearch.org/depts/biostats/software.