IANARS either, but no, it turns out not. Before you can drop into the sun you have to scrub off all the kinetic energy you have just from having been carried along by the Earth as it goes round the sun. Until then, gravity will no more pull your rocket in than it will pull the Earth in. And as long as you’re changing something’s kinetic energy, the figures show that it takes less of a change to pull out of the sun’s gravity well than it does to reduce it to zero (or as near as it takes to fall into the sun).
ETA: fie upon you, TroutMan!