  Exact null distributions of goodness-of-fit test statistics are generally
challenging to obtain in tractable forms. Practitioners are therefore usually
obliged to rely on asymptotic null distributions or Monte Carlo methods, either
in the form of a lookup table or carried out on demand, to apply a
goodness-of-fit test. There exist simple and useful transformations of several
classic goodness-of-fit test statistics that stabilize their exact-$n$ critical
values for varying sample sizes $n$. However, detail on the accuracy of these
and subsequent transformations in yielding exact $p$-values, or even deep
understanding on the derivation of several transformations, is still scarce
nowadays. The latter stabilization approach is explained and automated to (i)
expand its scope of applicability and (ii) yield upper-tail exact $p$-values,
as opposed to exact critical values for fixed significance levels. Improvements
on the stabilization accuracy of the exact null distributions of the
Kolmogorov-Smirnov, Cram\'er-von Mises, Anderson-Darling, Kuiper, and Watson
test statistics are shown. In addition, a parameter-dependent exact-$n$
stabilization for several novel statistics for testing uniformity on the
hypersphere of arbitrary dimension is provided. A data application in astronomy
illustrates the benefits of the advocated stabilization for quickly analyzing
small-to-moderate sequentially-measured samples.
