Creating a SLR model and performing a t-test on the slope parameter (or doing an F-test on the sum of squares) greatly depends on the sample size. I am dealing with very large sample sizes, so small R-squared values correspond to significance. Is there a way to test for correlation while controlling for small sample size?
There is no need to "control" for a large sample size, but it is always necessary to interpret the statistical results. If you have a very large sample size then a trivial correlation can give an impressively small P-value, as you know, but the correlation is nonetheless trivial. It is important to know that a large sample does NOT lead to a small P-value when there is no correlation (i.e. when the null hypothesis of no relationship is true).
Look at the correlation and decide if it is strong enough to be worthy of further attention. There is no statistical test for that, as it requires an understanding of the system being studied and no statistical procedure comes with such an understanding in-built.