Sounds like something is loose, perhaps the track bar mount is loose or its mounting bolt hole is wallowed out.
The steering stabilizer has nothing to do with DW, you should be able to drive without the SS in place without DW occurring.
Get a helper to turn the steering wheel back & forth with the tires on the ground while you look for any unwanted side-to-side movement on the steering components first, then the rest of the front suspension like the control arms and shocks.
Also an imperfectly balanced tire can be the root trigger of DW, as can hitting a bump or dip in the road. Then something that is loose allows the DW to develop from that triggering event.
Do check the toe-in which if it is out enough can encourage DW.