I suspect you mean Azure Maps and not Bing Maps. Bing maps is an older map platform that is being retired and doesn't have a SymbolLayer class.
With symbol layers it is possible to add some logic to drag symbols/text around however the performance isn't great since the symbol layer is rendered in the HTML canvas (WebGL) and each move creates a "repaint" of the map. Here is a sample of how to do this: https://samples.azuremaps.com/?sample=draggable-symbols
When it comes to draggable points on the map, the HtmlMarker class is much smoother since it's a simple CSS position change. You can try this out here for comparison: https://samples.azuremaps.com/?sample=draggable-html-marker
If you have a lot of labels being displayed, you will likely want to keep them rendered in the Symbol layer for performance, but to get smoother dragging, hide the selected symbol, then use an HTML marker to drag a copy of the symbol to a new position, then update the original symbol points coordinates in the data source and show it again.