Does cooking wine go bad like regular wine?
Generally... not as quickly. The salt content of the wine is much higher than drinking wine, which will preserve it longer. It will still "go bad" eventually. (Going bad is more a decrease in quality. It will still be edible, but unpleasant. It won't make you sick, except by the bad taste!)

See this from a Wikipedia Article: "Cooking wine typically available in North America is treated with salt as a preservative and food coloring."

So, it's safer than normal wine, but I'd try and use it soon. The sooner, the better.