You can use transform>warp to make you text "fit" the slices. Then you can create a shading layer out of the slices to adjust the text shading.
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A couple more options.
The first is with Photoshop. Use the Pen tool to create a work path that follows the required baseline of the text.
Select the Type tool and hover over the start of the path until you see this cursor, and start typing
If you happen to have access to Illustrator it has a trick where it can distort text into a shape using Object > Envelope Distort. LINK
Seeing as you're more savvy in Illustrator then why not do the distortions in that?
As you'll be aware they're better distortion controls in Illustrator