I received my Universal link cable yesterday evening! The good news is that I purchased the correct cable and I can transfer data to my Game Boy Printer. The bad news is that, as I suspected, the printing quality is terrible due to the aged thermal paper. I took a dark picture on my Game boy Camera to show how bad the problem is.
Unfortunately, Nintendo does not sell the thermal paper anymore (discontinued 2002-ish?). I did find a seller on Amazon that had the original Game Boy Printer paper. As I read the customers’ comments, I quickly discovered that the paper was no good. Despite it being in the original box it was initially sold in, the paper still aged and was used. Within the comments, though, someone suggested that receipt paper should work. The only problem with receipt paper is that it is too wide, but that is a problem that can be fixed.
So my next plan of action is to acquire some receipt paper to see how well it works with the Game Boy Printer.