Hold the nosewheel light as long as you are able, it's better for the nosewheel, and will make steering a little easier. Might you be unintentionally be applying brakes? Focus on keeping your heels on the floor, just as takeoff, until you need to apply the brakes.
Normally yes, for most light aircraft I agree fully, but the DR nosewheel at full extension is locked fore and aft. It needs
some (not a lot) of weight on the nosewheel before it will turn. It's aircraft specific, and the OP has been taught correctly. "Let it come down" BTW, the brakes need a good bit of rudder in a correctly set up aircraft before they come on. To brake both sides together involves pulling on a handle.
If it needs more than this, then there is a problem with the aircraft and it shouldn't be flying. You can see how much it needs by pulling down on the prop at some convenient point in the hangar or elsewhere. You may need to do this when pulling it out of the hangar anyway