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Constructing molecular classifiers that enable cells to recognize linear and non-linear input patterns would expand the biocomputational capabilities of engineered cells, thereby unlocking their potential in diagnostics and therapeutic applications. While several biomolecular classifier schemes have been designed, the effect of biological constraints such as resource limitation and competitive binding on the function of those classifiers has been left unexplored. Here, we first demonstrate the design of a sigma factor-based perceptron as a molecular classifier working on the principles of molecular sequestration between the sigma factor and its anti-sigma molecule. We then investigate how the output of the biomolecular perceptron,i.e., its response pattern or decision boundary, is affected by the competitive binding of sigma factors to a pool of shared and limited resources of core RNA polymerase. Finally, we reveal the influence of sharing limited resources on multi-layer perceptron neural networks and outline design principles that enable the construction of non-linear classifiers using sigma-based biomolecular neural networks in the presence of competitive resource-sharing effects.