Hello Nick and welcome to the site...
The thing is very simple.The chords continue ringing mostly because of the open strings that you play in a chord.Try playing The D chord from the open D string or even from the first E string.I guess that it will give you the effect you like and wish for...Also know that you wiil come across the same problem at the barre chords.Another thought is to try changing between chord more rapidly so that it wiil not be noticeable or better try holding your fingers down for a longer time so that the ringing effect that you say wiil go on...
Good luck with your practice....
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