correlation coefficient versus DEGs analysis: what's the best approach for low expressed genes?
First of all, RNASeq is extremely sensitive to batch effects.  Are these matched controls processed at the same time as the tumors?  IF they are not, lots of your differences will be caused by batch effect, and not cancer.

I believe the best strategy is to merge the FPKM columns and perform
correlation analyses of my gene of interest (GOI) versus all other
genes in both cancer and normal tissues.

What you propose isn't publishable.  Use the software that people have been using and validating for years.  Use either DESeq2 or EdgeR, or limma if you don't have access to raw gene counts.
If the underlying counts are low, the correlation won't mean much, because your real values could be quite off. If your gene has 2 counts in one sample, and 3 counts in another, that does not mean it really correlates with a gene that has 500 reads in the one sample and 750 in the other. I don't know that you can trick your way around it; you need more reads, or more samples for more power.