My sunchokes seem to getting bigger by the day, and may develop some rot if I'm not careful, so I was wondering how I should go about watering my sunchokes with "free" water to get the highest yield that I can possibly get.
Sunchokes are drought tolerant plants that prefer well drained soil. It would not be good to over water them in a poor draining soil.
There's no fixed periodicity to water them as it depends on your soil and weather. Unless you have indicator plants such as oleus plants (Coleus blumei) that droop when the soil is dry, you are either going to need to inspect the soil, or use a meter. Normally, you'd want to water deeply when they need it.