Can the police charge me with a breathalyzer refusal, even though I tried to blow into it?

Yes, you are required to give two adequate samples, pursuant to Florida law. If the officer makes a judgment call that you are not attempting to give him an adequate sample, him or her, then yes, they can charge you. However, that determination, that factual determination of whether or not you refused or not can be challenged at the administrative level in the Department of Highway and Safety Motor Vehicles process, and also challenged at the court level with a judge assigned to the DUI case.