Okay. This is going to be easy.
You're in the middle of a circuit and that's why you have 2 Romex'.
The HOT black is the line side. It comes from the panel. The not-hot black is the load side. These 2 blacks go to the brass screws on the switch. The whites tie together and don't go to the switch, UNLESS there are 2 silver screws on it, then they go there. Watch the switch to see if it's clearly marked for LINE and LOAD. In that, the HOT black goes to LINE and it's white companion (from the same Romex) goes to the LINE silver screw. The 2 from the second Romex go to the screws marked LOAD.
In the sconces, you're going to find one that has 2 sets of wires and the other has one set. (I'm betting) The one that's working better, is the first one from the switch wiring.
Open the working one with the switch turned off, of course. Disconnect the wiring going to the second light. Make sure the first light works just like you want it to.
Now, take the 2nd sconce down and see if the wires are in the right places. White to white, black to black, grounds in place, protecting wires from the others. If all that looks fine, reconnect the wire running from the first light to the second. Even though these are still opened up. turn on the switch and see if the 2nd sconce works.
If it's still buzzing, try to determine exactly where the noise is coming from.
Some dimmers won't work with florescent bulbs, some don't like halogen bulbs. SOME don't like the energy-efficient curly bulbs!
The box/instructions for the new dimmer may point out what it's NOT compatible with.
If everything's connected correctly and there's still buzzing, I'm guessing that there's some mis-match between the new dimmer and the lights.