  Given an affine rational complexity-one $T$-variety $X$, we construct an
explicit embedding of $X$ in affine space $\mathbb{A}^n$. We show that this
embedding is well-poised, that is, every initial ideal of $I_X$ is a prime
ideal, and determine the tropicalization of $X$. We then study valuations of
the coordinate ring $R_X$ of $X$ which respect the torus action, showing that
for full rank valuations, the natural generators of $R_X$ form a Khovanskii
basis. This allows us to determine Newton-Okounkov bodies of rational
projective complexity-one $T$-varieties, partially recovering (and
generalizing) results of Petersen. We apply our results to describe all
irreducible special fibers of $\mathbb{K}^*\times T$-equivariant degenerations
of rational projective complexity-one $T$-varieties, generalizing a results of
S\"u\ss{} and the first author.
