Hello. Thank you for coming to JustAnswer with your question, and welcome.
Fructose is basically fruit sugar. But, you are correct, cane sugar does have a fructose component. Some vegetables that are naturally sweet also contain a fair amount of fructose. Examples of these would be beets (both red and sugar beets), sweet peppers, sweet corn, sweet potatoes and yams. And there are many vegetables that contain fructose in small amounts Additionally, sucrose (another type of sugar) contains both fructose and glucose.
You will have to carefully check labels of premade foods very carefully. And even if you did eat a "fresh" diet with no foods prepared by anyone but yourself, you are still going to get small amounts of fructose in some of the vegetables you eat.
Here is a link to some information from MayoClinic's site: Fructose Intolerance
And, as this article suggests, I think it would be wise to consult a dietitian for some suggestions about how to maintain this diet, as it will be a bit challenging. You will need a complete list of foods that are OK to eat.
Here are some other links I found with some suggestions for foods that are "fructose free":
Fructose Free Foods
List of Foods with No Fructose Corn syrup (this list gives you an idea of what prepared foods you might buy that contain fructose as an additive, along with suggestions for some that do not).
Most breads and pastas are made with fructose, sucrose or corn syrup (high amount of fructose)....so would not be on your diet. Even some "unsweetened" cereals have corn syrup added.