After you pull out with the box, drive a bit, stop and check things out after a couple miles. 9 times out of ten, you are fine but...
+1!!
I can't count the number of times I've had to adjust and re-tighten ratchet straps after the first few miles, always with bulky odd shaped or asymmetrically weighted gear.
You might want to run a couple of straps circumstantially around the box to ensure the drawers stay closed, after applying moving blankets or cardboard. Then run one strap across the top, one across the front and one across the rear both of which make a U, ie the two hooks are at the rear and the apex of the U is the front of the box, then vice versa.
This way any sudden stops or acceleration will be halted with the front or rear straps, and any big bumps won't lift the box from the trailer and loosen the the other straps if you have one strap across the top.
This is how I strapped down two tanker desks on my wood floored landscape trailer, and I still had to tighten a couple of straps after a few initial miles of travel.