Of course you can give up sugar, and while you're at it, you should drastically reduce the total carbohydrate content of your Diet :)
Why? Because modern dietary advice and "foods" are killing us (a lot of what's in the supermarket barely qualifies as food imo).
Try googling "half of Americans pre-diabetic" (rates are a little lower in the UK but still appalling). Around half of adults in the US are either diabetic, or have metabolic syndrome/prediabetes, and even more shockingly, around 1 in 4 teenagers are pre-diabetic (this number should be close to Zero!). This is a huge, huge red flag.
On a modern diet, whole populations are fat and sick and heavily medicated. Our levels of chronic sickness are not natural, but result from a diet so far removed from real, unprocessed food that most people can't accept the concept that low sugar, lower carb, minimally processed diets aren't some sort of unnatural diet - it's REAL FOOD!
Try a little research, find some wonderful recipes, at these free blogs. MarksDailyApple, Robbwolf paleo solution blog, dietdoctor.com lchf (the food revolution).
It's true that we can't fully replicate a single perfect Palaeolithic diet, because this varied from season to season and geographical location, but we know what the were eating (real food!).
Meat, fish, eggs, nuts, berries, leaves, root veg...
They wouldn't recognise about 80% of the junk in supermarkets as being food.