Thanks for all the suggestions. I don't know how I'd survive without Techspot : )
Since I only had a handful of PDF files to merge, I used lopdog's recommendation of the online tool, MergePDF. That definitely worked to seemlessly merge my documents. However, now I am having trouble uploading them onto the Website where my application is being submitted. I don't know if it is just my slow dial-up service or problems with the Web site. In any case, thanks for the link. It definitely worked to merge my PDF files, without watermarks.
@mae: where do you get your pdf-files? If you create them yourself you probably have a free/commercial PDF printer. They often have an option called "Append" that lets you print a document into an existing .pdf file.
I scanned my documents from my Canon printer/scanner and saved them in PDF format. But, I could not save them all together in one document. At least, it appeared this was not an option.