Do extracts add sweetness or just the flavor?
I think you're conflating syrups and extracts.

Extracts are some flavoring oil plus alcohol.



For example, Vanilla Extract:


  Vanilla extract is a solution containing the flavor compound vanillin as the primary ingredient. Pure vanilla extract is made by macerating and percolating vanilla beans in a solution of ethyl alcohol and water. In the United States, in order for a vanilla extract to be called pure, the U.S. Food and Drug Administration requires that the solution contains a minimum of 35% alcohol and 100g of vanilla beans per litre (13.35 ounces per gallon).


There's no sugar in extracts, it's just flavoring... very concentrated flavoring... with a bit of alcohol. It generally comes in very small bottles though large bottles are available for commercial uses.

If you want sweetness and the flavor, try flavored syrups (like they use in coffee bars to make lattes)... but only in beverages and other similar recipes.  You can't use these instead of sugar and extract in baking.