We propose a p-adaptive quadrature-free discontinuous Galerkin method for the shallow water equations based on a computationally efficient adaptivity indicator which works without any problem-dependent parameters. The error and smoothness of the solution are detected using the information collected for slope limiting and, for piecewise constant discretizations, by carrying out a reconstruction procedure. The accuracy and robustness of the new scheme are evaluated using several benchmarks and compared to other adaptivity indicators. Our results indicate that the proposed indicator finds a good balance between solution quality and computational overhead.