I did not get many responses over how to compute $p$ values given Spearman $\rho$ and Pearson $r$ so I will try something else:
I see that when you are using the Spearman $\rho$ value, you can use this to determine critical values depending on $n$:
However I also see this for Pearson's $r$:
How would I compute these directly (given arbitrary significance level, degrees of freedom, $r$, etc) instead of relying on a pre-computed table?
I see absolutely nothing online illustrating how this is done. I am comparing my results against these calculators (which work for small sample sizes): www.socscistatistics.com/tests/