We study interacting Rashba-Dresselhaus fermions in two spatial dimensions. First, we present an exact solution to the two-particle bound-state problem of spin-orbit (SO)-coupled fermions for arbitrary Rashba and Dresselhaus SO interactions. An exact molecular-wave function and the Green's function are derived explicitly along with the binding energy and the spectrum of the molecular state. In the second part, we consider a thermal Boltzmann gas of fermionic molecules and compute the time-of-flight (TOF) velocity and spin distributions for a single fermion in the gas. We show that the signatures of the bound state can be observed already in the first-moment expectation values, such as TOF density and spin profiles.