Local discontinuous Galerkin (LDG) methods are in regular use in literature and industry to model conservation law type problems that contain spatial derivatives of higher order than one; such problems may often exhibit stiffness. Implicit-explicit (IMEX) time-stepping methods have also seen common use to efficiently solve problems which may have both stiff and non-stiff components. Only recently has work begun in the application of IMEX methods in conjunction with LDG methods to solve such problems. In this work we are particularly concerned with such an application with IMEX Runge-Kutta (RK) methods. We initially repeat recent error convergence and stability results by Wang, Shu and Zhang for a one-dimensional (1-D) convection-diffusion problem with LDG discretization in space and IMEX Runge-Kutta (RK) discretization in time. We also achieve new corresponding results for a likewise discretized two-dimensional (2-D) linearized shallow water problem, in which a constant eddy viscosity term introduces stiffness to the problem. Both our 1-D and 2-D problems are modeled inefficiently by purely explicit methods, with strict time-step restrictions imposed on each in this case, due to their stiffness. Using IMEX methods, one observes optimal error convergence rates as well as relaxed restrictions on time-step sizes in both problems. We present such results as well as additional experimental results such as comparisons of computational run-times and maximal time-steps for the purely explicit and IMEX cases on both types of problems with varying degrees of stiffness. We conclude that IMEX RK methods are more consistently efficient than the more commonly used standard explicit strong-stability preserving RK methods for the solution of stiff problems. We observe the relationship of efficiency and time-step improvements to the ratio of the degree of stiffness to non-stiffness of a problem in both one and two dimensions, similarly to results for maximal stable time-steps obtained by Wang, et. al. in 1-D.