Will furrow irrigation save more water than flooding my whole field?

Yes, for row crops, furrow irrigation usually saves water compared with flooding the whole field, because you water the furrows between the ridges instead of covering the entire surface. The plants on the ridge draw water from the wet furrow, and less of the surface is exposed to evaporation. You can save even more with alternate furrow irrigation, where you water every second furrow each time and switch the next time, which roughly cuts water use while keeping the crop supplied, and it suits wide spaced crops on lighter water. Furrow works best on a field that is reasonably level, so the water runs evenly to the end rather than pooling at one spot, which is another reason land leveling helps. For very high efficiency, drip beats furrow, but furrow is a cheap and easy first step away from flooding. Whether alternate furrow suits your crop and soil depends on the crop and your conditions, so confirm with your KVK before switching on a large area.