Here are the before and after pictures

view in Finale

View in PDF reader after printing to PDF.  I've tried both Adobe and Nuance Power PDF

How are you creating the PDF? Finale’s output has been flaky for quite some time. You might try exporting through the Graphics tool, or use one of the free PDF printers available for Windows.

I am using Bullzip PDF Printer.

Try the graphics tool. Are the underlined text inserts hyperlinks?

It looks the same if I print using the Export to PDF. 

How do you export using the graphics tool?

Wait, if I export it as a graphic, would I then convert the image to a PDF?

Okay, I just exported it through the graphics tool, and PDF was one of the options.  But it still looks the same, with the underline too close.

Oh, and yes, the underlined texts are hyperlinks.  But since the links don't work for some reason, I add them in Power PDF afterwards.

That may be the cause. I'd send it in to tech support. Use the Submit a Request tab at the top of the page.

Can you enter the hyperlink text after you generate the PDF? Just leave it out of the Finale file?

I took away the hyperlink and just underlined the text in Finale.  Same thing happened.  It's an underline problem.

When I add the underline to the text in Adobe Reader, it's all fine. I take the underline away from the hyperlink and add it back in. 

Don't underline the text in Finale. You could even leave it out, and just add it in as a hyperlink in the PDF. I'm assuming you are the only one with access to the Finale file, and you would only distribute the PDF.

What font and size are you using? Some fonts and sizes do not behave when saved as PDF.

A PDF file can be a picture of a page or it can be a set of instructions using a set of embedded fonts—that is what is happening here.

I suggest trying Minion Pro. It is sandboxed in every Adobe product, even the free Reader, but there are ways to make it available to all applications in Mac & Windows. Being an Adobe font, this should help pinpoint the problem.