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Molecular computation involving promiscuous, or non-orthogonal, binding interactions between system components is found commonly in natural biological systems, as well as some proposed human-made molecular computers. Such systems are characterized by the fact that each computational unit, such as a domain within a DNA strand, may bind to several different partners with distinct, prescribed binding strengths. Unfortunately, implementing systems of molecular computation that incorporate non-orthogonal binding is difficult, because researchers lack a robust, general-purpose method for designing molecules with this type of behavior. In this work, we describe and demonstrate a process for the rational design of DNA sequences with prescribed non-orthogonal binding behavior. This process makes use of a model that represents large sets of non-orthogonal DNA sequences using fixed-length binary strings, and estimates the differential binding affinity between pairs of sequences through the Hamming distance between their corresponding binary strings. The real-world applicability of this model is supported by simulations and some experimental data. We then select two previously described systems of molecular computation involving non-orthogonal interactions, and apply our sequence design process to implement them using DNA strand displacement. Our simulated results on these two systems demonstrate both digital and analog computation. We hope that this work motivates the development and implementation of new computational paradigms based on non-orthogonal binding.