How can I get my 5 a day without exceeding 24g of sugar?

I am trying to eat healthily and break my habit of craving for sugar. At the moment I am following a diet plan for which I don't exceed 24g of sugar. I have been managing this fairly easily, but only because I've been cutting down on my fruit and vegetable intake. Fruit is so sugary that I can't eat too much, but annoyingly, all the vegetables that are good for snacking on - such as carrots - are really sugary. I feel a bit minging and unhealthy for not eating lots of fresh stuff. Is there anything I can do, or will I just have to accept that I'll have to eat more sugar?