The ship is also painted with a radar reflective X that the booster is targeting on the way down. The ship just has massive engines for solid station keeping in rough seas. Like you say they don't use those to move to the booster.

I'm fairly sure it's these things: https://www.thrustmaster.net/out-drive-propulsion-unit/

They were posted about when the ships were first made. And I know spacex had gone a size up from the originals.