We consider finite element approximations of the Oldroyd-B equations under no flow boundary conditions in a bounded domain. We use continuous piecewise quadratics for the velocity field and either (a) piecewise constants or (b) continuous piecewise linears for the pressure and the symmetric conformation tensor. Our schemes satisfy a free energy bound without any constraint on the time step for the (backward Euler) time discretization. This extends results of [Boyaval et al. M2AN 43 (2009) 523--561]: there, one required our approximation (a) to treat the advection term in the stress equation plus a restriction on the time step. Furthermore, for our approximation (b) plus an additional dissipative term in the stress equation and a cut-off on the conformation tensor (like in [Barrett et al. M3AS 18 (2008) 935--971] for the FENE equations), we show (subsequence) convergence, as the spatial and temporal discretization parameters tend to zero, towards global-in-time weak solutions of this regularized Oldroyd-B system. In the 2-dimensional case we also carry out this convergence with a time step restriction dependent on the spatial discretization, but without cut-offs.