Here's how I would do it.
Create your two blend objects. Copy the top shape to the clipboard.
Create a blend using the maximum number of steps (999).
Convert the blend to Editable Shapes (Arrange menu) then ungroup.
With all 999 steps selected, Arrange > Combine Shapes > Add Shapes. This may take some time. But it will create one outline.
Paste the text on top (Edit > Paste in Place).
Use the Contour Tool to apply a one-step Inner Contour to the width of the text outline.
Convert to editable shapes and ungroup.
Apply a white fill to the inner contour shape.
That said, the way I would really do it, which is more accurate and produces a cleaner vector image, is to select the two text shapes and Add Shapes. (You still need to copy the top text to the clipboard first).
Switch to the Shape Tool and delete the points shown in my attachment to make the edges.
Paste the text copy on top from the clipboard and apply the same contour as outlined in my first example.
This produces a very clean drop vector shadow that will not make the plotter cry out in pain.
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