We present stable and convergent mixed finite element methods for the discretisation of poroelasticity and elasticity/poroelasticity coupled equations in a new formulation, where the volumetric contributions to the total stress are merged into an additional unknown. The resulting saddle point formulations can be analysed by means of a Fredholm alternative, after realising that they are compact perturbations of Stokes-like invertible systems. Galerkin schemes are then proposed, and given that specific finite dimensional spaces are chosen adequately, the methods remain stable and optimally convergent in the incompressible limit.