If you're using InDesign CS6, the Auto-Size feature is now built in. It's in the Object > Text Frame Options > Auto Size tab.
I tried the Auto-Size feature with a little CSV file and Data Merge and it seems to be working on a very simple one-per-page merge. I'm using the Preview feature to preview the records. See the pictures below:
Here are the settings in the Text Frame Options dialog:
I'll let you try it out further to see if you have the same misalignment problems as you found with TypeFi auto-fit.
But I wouldn't be too optimistic. I'm pretty sure there were no changes to the Data Merge feature in recent versions to account for variable frame size.
I tried that after I posted, and got the same result. Data Merge just can't handle it.
Auto-size was built with apps in mind for liquid layout, and I don't think it was built for this purpose. Similarly, if placing inline graphics into auto-size frames that use fit frame to images, the merge works 1-up but absolutely fouls up on multiple records per page.
If all that is being done is merging text to ultimately merge into one large textbox, Loic Aigon made a solution that received some good press via InDesignSecrets: http://indesignsecrets.com/data-merge-into-inline-anchored-objects-so-they-flow-in-a-story .php
The article's title is misleading, but ultimately his script will do the impossible - it will merge many data records within the one text frame. This eliminates the need in this instance for dozens of textboxes, and the text can flow in one story.
I had the same problem as posted here. In the end, what worked for me was rebuilding and rethinking the layout so that the data merge content went into table cells in a table rather than auto-fit frames and anchored objects in a story.
Table cells seem to work fine with data merge content, resizing exactly as expected with none of the weird bouncing around or overlapping as seen here. They're not the easiest things to work with, but there's a lot of control in there.