Using Noteworthy Composer, instead of inserting the chord names above the music as text fields, insert them below the music as lyrics. This will mean that you need to mark most of the notes as "Lyric syllable: Never". The easiest way to do that is to select the entire melody, right click it, select Properties, select the Notes tab, and set the Lyric Syllable property for all notes to Never. You can then mark selected notes to have lyrics syllables (which are, in your case, really chord names).

Save the file as a Type I MIDI file as well as saving it as a .nwc file. Using Van Basco's Karaoke player, as suggested above, will let you display the lyrics (chord names) in your MIDI file without converting it to a .kar file. I don't know what other Karaoke players have that nice feature.

Note that there seems to be a bug in one or the other program that, in my test file, prevented the first note's lyric syllable (chord name) from displaying. I worked around that by putting a rest and a bar line in front of the beginning of the real melody.