Let W be a finite Weyl group and (W) over tilde the corresponding affine Weyl group. A random element of (W) over tilde can be obtained as a reduced random walk on the alcoves of (W) over tilde. By a theorem of Lam (Ann. Prob. 2015), such a walk almost surely approaches one of vertical bar W vertical bar many directions. We compute these directions when W is B-n, C-n, and D-n, and the random walk is weighted by Kac and dual Kac labels. This settles Lam's questions for types B and C in the affirmative and for type D in the negative. The main tool is a combinatorial two row model for a totally asymmetric simple exclusion process (TASEP) called the D*-TASEP, with four parameters. By specializing the parameters in different ways, we obtain TASEPs for each of the Weyl groups mentioned above. Computing certain correlations in these TASEPs gives the desired limiting directions.