We introduce a class of adaptive timestepping strategies for stochastic differential equations with non-Lipschitz drift coefficients. These strategies work by controlling potential unbounded growth in solutions of a numerical scheme due to the drift. We prove that the Euler-Maruyama scheme with an adaptive timestepping strategy in this class is strongly convergent. Specific strategies falling into this class are presented and demonstrated on a selection of numerical test problems. We observe that this approach is broadly applicable, can provide more dynamically accurate solutions than a drift-tamed scheme with fixed stepsize, and can improve MLMC simulations.