Make sure music is not playing, and that playlists are showing. Long-touch the playlist icon, "enqueue", repeat until stuffed, press play.
"Walkman" has great playback for WAV files, but it lacks many functions and finesse. Here's how I've made do with when not using VLC:
1st - All music is stored "per album" or per "playlist", each of these are copied to their own folder. I prefer running dupe file than trusting crappy "music classification", because types are usually messed-up. Case and point: Metallica set as "classical" on one popular CD database....
2nd - If you use local storage and SD card storage for your music, organize everything in neat folders.
3rd - Run
My playlist. (Free).
4th - Run Gracenote to collect music data, allow re-writing.
5th - Check the playlist names, specially for albums. Some names are set to start with "millenium collection series.. etc", so you don't know what the songs's name, there just not enough display room.
6th - Make sure music is NOT playing, got to playlists, long-press chosen playlist, "enqueue", repeat until all is chosen, go to "now playing" tab, then press play.
7th - Write Email to Sony customer care and tell them it's a real shame that a legendary name like "Walkman", the embodiment of portable music, is given to a piece of software so incomplete. We need a folder viewer and browser, and a playlist editor that supports "playlist by folder".
Have a good one.