Is the playlist in your iTunes library, or is it already on the shuffle (as a playlist)?
If you created one or more playlists for your shuffle in your iTunes library (with the songs in the desired playlist order), you should set up automatic syncing. Here's how to do that, if you have not already...
NOTE: If the iTunes sidebar (along left side of window) is not visible, from the menu bar, under View, select Show Sidebar.
Select the shuffle in the sidebar, under DEVICES. Near the top of the window, there is a bar of buttons, starting with Summary. Click Music next to Summary. This is the shuffle's Music screen, where you tell iTunes how to sync songs. (Sorry if you know this already...)
Check the box for Sync Music. Below that, choose the option to sync Selected playlists, artists, albums, and genres. In your case, find and checkmark those playlists you want under Playlists. When you click Apply, the songs on those playlists and the playlists sync to the shuffle.
When using your shuffle, the default playlist that you are using is the All Songs list. To play songs in order, the shuffle's power switch needs to be set to the play-in-order (middle) position. However, if you are playing from the All Songs list, the songs play in alphabetical order.
To play from a particular playlist that is on the shuffle, you need to use VoiceOver to switch to that playlist. This document explains how to set up and use VoiceOver on a shuffle
When playing from a particular playlist, with the power switch set to play-in-order, the songs play in playlist order.
Glad these old post still help... If you're using iTunes 12, good job following what I wrote; previous versions of iTunes were somewhat different. In case anyone else reads this topic, here's updated info for iTunes 12.
Select the shuffle in the sidebar, under DEVICES.
Instead of selecting the iPod in the sidebar, find the iPod's "device button" on the horizontal bar that goes across the iTunes window, and click it. The iPod's Summary settings screen is shown. In the sidebar of that screen, under Settings, click Music. This is the iPod's Music settings screen, where you tell iTunes how to sync songs.
The rest of the previous old post is still valid.