Overlapping text with InDesign's data merge

I'm trying to set up a book of song lyrics with a two-column layout. I'd like to pack as many songs as a page as will fit, so I tried doing a data merge with multiple records per page.

This works great, but wastes space because it forces only one song per column:

Since each song has a different length, I tried making the lyrics section a variable height using Auto-Sizing:

But when I do that, the resulting document overlaps a lot:

Is there a way to use auto-sizing height but have it not overlap?