I used Greyhound for oddball oversized shipping several times. They just ended that service, though. Trailways seems to offer a similar one.

When I worked with LTL carriers in my retail days, they didn't require pallets or shrinkwrapping - just clearly labeled boxes and "x of (total)" boxes on each label. (It's been ages though.)

I've heard about (but never tried) some movers that will sell extra space on a van. Seems Seattle to Houston might be a fairly common move.

Good luck!