A sensitivity analysis in an observational study tests whether the qualitative conclusions of an analysis would change if we were to allow for the possibility of limited bias due to confounding. The design sensitivity of a hypothesis test quantifies the asymptotic performance of the test in a sensitivity analysis against a particular alternative. We propose a new, non-asymptotic, distribution-free test, the uniform general signed rank test, for observational studies with paired data, and examine its performance under Rosenbaum's sensitivity analysis model. Our test can be viewed as adaptively choosing from among a large underlying family of signed rank tests, and we show that the uniform test achieves design sensitivity equal to the maximum design sensitivity over the underlying family of signed rank tests. Our test thus achieves superior design sensitivity, indicating it will perform well in sensitivity analyses on large samples. We support this conclusion with simulations and a data example, showing that the advantages of our test extend to moderate sample sizes as well.