We used Amtrak Express shipping for our cross country move. Similar setup, no furniture just a bunch of boxes. It was the cheapest way to move that we found. 100lbs for $60 then 0.57 for each additional lb. Books we shipped USPS media mail as that rate is a little cheaper but everything in the box must be of the appropriate type. We then drove cross country and it actually beat us there by a day(they hold it up to 72hrs free then start charging). Only ideal of course if both your origin and destination have a station nearby and you will need to either deliver/pickup or hire someone to do so.
This article covers the process pretty well.
How I moved across the country with Amtrak - Transitized
https://www.amtrak.com/express-shipping
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