As long as it stays warm the plant will keep putting out tubers. That's not to say it will increase your yield as older can and will die off, rot and feed the worms. You will be able to tell the difference between the older tubers and the newer ones as the older tubers color seems to fade after about 12 months in the ground.

You should be fine, I grow turmeric and ginger in my house in Massachusetts. when I want some I pull up the entire plant, clip off some of the newer growth and dispose of the older growth, then put it back and give it a month or two to "heal" and regrow. Just make sure not to over-water it, it likes warm moist soil but too much water can give it root rot. Also leave some of the new growth to re-establish to replace the old that dies off.

The roots get "woody" and hard to cook with, and some may wither.

Do you have room to have several plants, each planted at a different time for a progression of harvests. When you dig up the first one, replant some tubers. When you dig up the next one, replant some of the tubers.