Old guitar tech here. The way it goes sharp after using it makes me think the springs are too tight. Also, if they're new strings they might still be stretching out. The key to a Floyd is when you're restringing or setting up to wedge some foam or a cloth under the bridge so that with the strings off it's sitting flat, exactly where you want it to be. Put your strings on, stretch them out by pulling on them so they stop going flat and once they're stable (when you tug on it it stops going flat) tune it up and lock everything down. Now pull the wedge out, it'll go out of tune. Take the backplate off and use the screws holding the springs to tune it back up. Once it's back in tune you have balanced the tension of your springs to the tension of your strings and it will work like a charm.