Towing without equalizer bars hooked up

I have, I don't see any reason no too as long as everything is good and tight (it should be!). I generally don't though, since the tt/van drop a few inches if I don't have the bars on. If I'm going to tow it any distance (like more than from the side of the house to the street) without the weight distribution bars in place, I use a different ballmount that is a few inches higher so the trailer still sits level after hooking up.

What is a "stinger"?

You can tow that way but will not have any of the advantages of the WD hitch. You didn't mention the brand of hitch but some have built in sway control with the bars in place.

If your TT is balanced right, i.e. sufficient tongue weight, some maintain you don't need sway control. I tend to not agree. I think sway control is a safety factor. The biggest disadvantage might just be unweighting your front axle and the resulting understeer