i think terminology is muddled here what you discribe is OPEN session the disk is not closed each time you add new files so cannot be read in any other device usually other than a BURNER/CD WRITER
A multi session disk is closed after each writing session until the disk is full and no more data can be added
Is this what you are telling us !?
try looking at a disk that does work fully in your computer explore the disks contents and layout how the files and folders are arranged on the disk and make a note of this to use as reference when making your next disk
also check using winamp or other windows mp3 player as to what tag is used either ID3 or ID2 as this may or maynot make a difference to the disk !?
At the same time check the files for length in characters (make a note of this too) and if there are any symbol characters in the name such as .. ()"£$%^&*._-=# or other !? this may affect the abiltity to play also as some players do not support those characters and often are limited to file length of name some only 8 characters
write the disk as a multi session using NERO or any multi session capable burner that supports burnign of MP3'S
come back here and let us know your results