I have tracked down some results on explicit classifications of simple modules for $u_q(\mathfrak{sl}_3)$. The general picture is that the simple modules are bigraded by the root lattice and look like towers of concentric hexagons.
For the benefit of anyone else interested, there is a long series of papers by Dobrev:
Multiplet classification of highest weight modules over quantum universal enveloping algebras: the Uq(SL(3,C)) example in Groups St Andrews 1989 vol 1 LMS LNM #159
Representations of Quantum Groups, Symmetries in Science V (Lochau 1990), 93–135, Plenum Press, NY, 1991.
A chapter from Lecture Notes in Physics, 1990, Volume 370, here
Dobrev-Truini Irregular Uq(sl(3)) representations at roots of unity via Gel’fand–(Weyl)–Zetlin basis
Dobrev-Truini Polynomial realization of the Uq(sl(3)) Gel’fand–(Weyl)–Zetlin basis
...and many others.
also there is a paper by Abdesselam, Arnaudon, Chakrabarti and a discussion of dimensions by Pereira here
Some of the relevant material is hard to find and/or requires paying large sums of money to publishing corporations.