Asked to create data set where magnitude of paired t-test statistic is very large, but pooled t-test statistic is small. Can't find any resources It seems almost impossible to make these tests diverge by much. I have tried making the variance for one distribution much larger, or much smaller. I have found a data set where the paired is a good bit higher for paired, but I'm not sure why...

I assume you want an example where there is paired data, the paired t test is highly significant, but if one erroneously use an independent samples t test that does not seem significant. Write first a model for paired data:
$$
  y_{iA}=\mu +\alpha_i + \epsilon_{iA}, \\
  y_{iB}=\mu +\alpha_i + \beta_B + \epsilon_{iB}
$$ where $\alpha_i$ is a random subject effect, common for the two treatments $A, B$, $\beta_B$ is treatment effect and $\epsilon_{i,A(B)$ is the error term.
As this looks like self-study, I will only hint. If you erroneously analyze using an independent samples t-test, the subject effect $\alpha_i$ will be thrown into the error term, making it difficult to see a treatment effect. So now you can simulate this in R (or some other software), vary the variance of $\alpha_i$ (and other terms), and experiment ...
