We obtain asymptotic formulae for the number of solutions to systems of inhomogeneous linear Diophantine inequalities over the field of formal Laurent series with coefficients from a finite fields, which are valid for almost every such system. Here `almost every' is with respect to Haar measure of the coefficients of the homogeneous part when the number of variables is at least two (singly metric case), and with respect to the Haar measure of all coefficients for any number of variables (doubly metric case). As consequences, we derive zero-one laws in the spirit of the Khintchine-Groshev Theorem and zero-infinity laws for Hausdorff measure in the spirit of Jarník's Theorem. The latter result depends on extending a recently developed slicing technique of Beresnevich and Velani to the present setup.