You are experiencing a NORMAL, temporary reaction that many people get. It means the Botox shot should work and give you a near normal voice when the breathy voice goes away. How long will the breathy voice last? Sorry but that can be different for each person. It could last days or even weeks. You can minimize the breathiness by lowering the dosage.
Getting a shot into only one side can also help lessen the breathiness.
The tickle in your throat should go away within a week of getting the shot. In the meantime, you can eliminate the tickle by bending your head down, lowering your chin down to your chest before swallowing. Try it, it should work.