You can set the Billboard's 'eyeOffset' to closer Z, however it might clip the model. If it looks bad, maybe you can shift the billboard relative to the center, but in screen coordinates - using 'pixelOffset' on the billboard.
var entity = viewer.entities.add({ name : <NAME>, position : position, orientation : orientation, model : { <YOUR MODEL> }, billboard : { image : 'YOUR IMAGE', eyeOffset : new Cesium.Cartesian3(0.0,0.0,-100.0), // Negative Z will make it closer to the camera
pixelOffset : new Cesium.Cartesian2(0.0,-32.0) // Optional offset in pixels relative to center } });
There was a post explaining how to add a line from a billboard to the ground, maybe it can help.
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