Is it copyright infringement if all the words to a song are changed but the melody remains unaltered?

Your idea is quite common. In fact, Weird Al Yankovich has made a very good career at changing the lyrics to songs while maintaining the melody and then performing the altered song. Although his humorous changes are clearly parodies of the original songs [or of the performing artists] and likely do NOT infringe the song's copyright, he always seeks a license from the owner of the copyright in the musical composition [i.e., the songwriter]. Weird Al's approach is best -- that is, you need to acquire a license to record and/or perform the song with the altered lyrics. The fact that the altered song would be performed at a "private" function -- and I don't know what that means -- is irrelevant [unless the function is one of a handful that are (essentially) statutorily immune from copyright infringement such as certain religious events, agricultural and horticultural fairs, veterans events, political events, events for the handicapped, etc]. As for your book, you can lawfully publish your altered lyrics and can lawfully suggest that they be used with a particular song. That is not an infringement of the copyright in the lyrics [unless you change them only slightly] or an infringement of the melody. Good luck.