Here's one way:
You can also try using the actual dodge and burn tools on the new layer instead of the brush tool, which might give you better results.
Also, try the Soft Light blend mode
One way to dodge and burn on a separate layer is to add a 50% gray layer as a new top layer set it blending mode to overlay. The image should look like it did before adding the new top dodge and burn layer. Next paint on this dodge and burn layer with low opacity black and white paint. White paint will dodge and black paint will burn. Painting with 50% gray will remove dodge and burn effects. You may also be able to use the burn and dodge tools on the layer targeting highlights and shadow areas. I just use black and white paint.