How much free space on your internal drive and how much free space on the Time Machine drive? You can use Disk Utility in Applications/Utilities to find free space. If Time Machine gets too full it could delete your backups. I suggest your turn off Time Machine in System Preferences until you get your files restored.
Rather than enter Time Machine using the Time Machine interface, go to the extrernal drive and drill down in the backups.backupdb folder. Go to a date when you know the files existed. Select Macintosh HD > Users > You > Pictures. Do you see the data. Check size? Do the same for Music.
You can drag from these folders to your Desktop as long as you have space on your internal drive.
I suspect you'll need to erase your Time Machine and start new or you'll need to purchase a larger drive. It's recommended that your Time Machine drive is 3X larger than internal drive.