I agree with S10 GXN , a full inspection of the front end is needed.
After any worn parts are replaced, then get an alignment. And not just a "set the toe " kind either. Thats a lazy mans algnment.
I suspect the added weight of the wrecker body and all has caused the rear to ride lower, which changes the caster angle,probably enough to cause a wobble that you are describing.
Next time you go to the grocery store ( for beer no doubt) watch some of the shopping carts as the people are pushing them. You will see one every now and then that has a front wheel that wobbles back and forth like you are describing. This is due to excessive caster. Hmmm maybe that is why those little front wheels are called CASTERS. Peace