Classic alignment algorithms utilize scoring functions which maximize similarity or minimize edit distances. These scoring functions account for both insertion-deletion (indel) and substitution events. In contrast, alignments based on stochastic models aim to explicitly describe the evolutionary dynamics of sequences by inferring relevant probabilistic parameters from input sequences. Despite advances in stochastic modeling during the last two decades, scoring-based methods are still dominant, partially due to slow running times of probabilistic approaches. Alignment inference using stochastic models involves estimating the probability of events, such as the insertion or deletion of a specific number of characters. In this work, we present SimBa-SAl, a simulation-based approach to statistical alignment inference, which relies on an explicit continuous Markov process for both indels and substitutions. SimBa-SAl has several advantages. First, using simulations, it decouples the estimation of event probabilities from the inference stage, which allows the introduction of accelerations to the alignment inference procedure. Second, it is general and can accommodate various stochastic models of indel formation. Finally, it allows computing the maximum-likelihood alignment, the probability of a given pair of sequences integrated over all possible alignments, and sampling alternative alignments according to their probability. We first show that SimBa-SAl allows accurate estimation of parameters of the long-indel model previously developed by Miklós, Lunter and Holmes in 2004. We next show that SimBa-SAl is more accurate than previously developed pairwise alignment algorithms, when analyzing simulated as well as empirical datasets. Finally, we study the goodness-of-fit of the long-indel and TKF91 models. We show that while the long-indel model fits the datasets better than TKF91, there is still room for improvement concerning the realistic modeling of evolutionary sequence dynamics.