First of all, we have to realize that the price of water is fixed in most markets. Governments set a price for water and that price does not change. This means that drought would not affect prices. I will, however, answer this question assuming that the price of water is set by supply and demand.
As your question tells us, the supply of water tends to be rather fixed. Therefore, it can be graphed as a vertical line. This implies that only the demand for water can move. So what happens when people are encouraged to conserve water? What happens is that they use less water (assuming they are actually practicing conservation techniques). If they are using less water, then demand for water is dropping. This is shown by a movement of the demand curve down and to the left.
If the demand curve goes down and the supply curve is vertical, we end up with a drop in the price of water. A drop in the price of water will encourage people to use more water. This will work to defeat the purpose of conservation. As people conserve water, the price drops, encouraging them to use more water. This implies that it will be hard for conservation policies to work if the price of water is free to change with supply and demand.