Can I sink existing concrete?
In your shoes, I'd do this:


Trench out along the concrete strip to give yourself access to the soil under the raised portions of the strip. Make the trench 6-8" below the bottom of the concrete and wide enough to work in. 
Determine whether there's remaining root under the concrete. If so, dig and chop it out completely.
Using a water hose, begin "drilling" under the high spots in the slab, allowing the water to wash soil out into the trench. Do this every foot or so. Rather than completely removing the soil in any large areas, remove it from several small areas. This will result in more predictable settling.
With a heavy object padded with towels or plywood, tamp the concrete down as you flush soil out from underneath.


Some notes:

Be sure to get the hose all the way to the far side of the concrete strip to accomplish a uniform descent and prevent tilting.

Stop when the slab is about 1/2" above the adjacent concrete. Chances are the slab will settle further as the soil fills the holes back in over time. Do occasional tamping over the next day or two and see where it stabilizes. Remove more soil if necessary, down to 1/8" or so above the adjacent slab.