When I first started shooting, I had a Nikon D5000, and I remember the first time that I took a shot of the moon on a particularly brilliant full moon night. Not to mention, there was a helicopter passing by the moon at the time of the picture. I had no idea what I was doing, but I'd be lying if I said there was no happy dance involved. So I'd like to think that I understand what you're feeling right now. It's not about a desired effect, like some people here are going on about. It's more about what YOU think works. The rest will come with time (photo stacking, star trails, intervelometers, cable releases, etc.)

What you should focus on right now is sticking with shutter priority. Absolutely get yourself a tripod. Walk around your neighborhood after the sun has set. Set your ISO way low (no higher than 400) on 20-30" shutter speeds. Your aperture should set itself around F/8 to 16.

One problem that you're going to inevitably run into when photographing the night sky, especially in urban areas, is light pollution. In the first shot that you included with your D5600, notice how the building with the solar panel is nothing more than a silhouette. The longer you leave the shutter open, the more you run the risk of blowing out the highlights. (that is, making the bright areas too bright). In other words, taking shots of the night sky is exceptionally challenging if you're not out in the 'sticks'. (Keep in mind that the same principle applies to the moon, as it's a source of light too.)

tl;dr practice practice practice (on shutter priority) with longer shutter speeds and low ISO. get a tripod.