Affleck-Dine (AD) baryogenesis along a d=6 flat direction in gauge-mediated supersymmetry-breaking (GMSB) models can produce unstable Q-balls which naturally have field strength similar to the messenger scale. In this case a new kind of Q-ball is formed, intermediate between gravity-mediated and gauge-mediated type. We study in detail these new Q-ball solutions, showing how their properties interpolate between standard gravity-mediated and gauge-mediated Q-balls as the AD field becomes larger than the messenger scale. It is shown that E/Q for the Q-balls can be greater than the nucleon mass but less than the MSSM-LSP mass, leading to Q-ball decay directly to Standard Model fermions with no MSSM-LSP production. More significantly, if E/Q is greater than the MSSM-LSP mass, decaying Q-balls can provide a natural source of non-thermal MSSM-LSPs, which can subsequently decay to gravitino dark matter without violating nucleosynthesis constraints. The model therefore provides a minimal scenario for baryogenesis and gravitino dark matter in the gauge-mediated MSSM, requiring no new fields.