AP, my preference is to do just like you said. However, now I'm re-evaluating in certain circumstances. For instance, one dealer gave me a much better deal than another but I still went back to the original dealer to give him a chance to match. But there's no way the first dealer would have given me that price had he not believed he was going to lose the sale. Now I think l#ll give it to the dealership who was willing to offer me the price in the first place, unless I had taken up significantly more time with the first dealership than a simple test drive. I try not to Easter people's time unless I'm ready to start negotiating, so I also like to be very upfront with, including if I will seek another quote elsewhere, usually at a place recommend by a friend or has a good reputation. I have not asked for quotes from multiple dealerships.
Edit: sorry for typos, writing from Android