I have an Exmark Viking, 36" hydro driven w/b. I have had it for a long time, and used it without sulky, with a one-wheeled Velke, and with a two-wheeled Bull Rider (convertible model).
The one-wheeled sulky is highly maneuverable, easy on/off, lightweight. But, I did not like the line down the middle of the mowing pass.
I bought the Bull Rider, intending to use it is caster mode (what I think you are calling swivel). Each of the two caster wheels have spindles/yokes that can be free to turn, or locked down. When locked down, a pin in the hitch is pulled out, permitting the tow hitch to pivot, making the sulky a trailer model.
My intention was to use the BR in caster mode, but quickly realized it would not work. The mower is too light. The sulky drives the mower, rather than the mower driving the sulky. Power isn't a problem, nor traction. But, it is the direction control that was a problem. Also, the mower is very light in the front already, and the sulky made the problem worse. The footprint of the 36" model is simply too narrow to have sufficient leverage to make the caster mode work properly. I could not make turns like I wanted, nor could I feather the controls to drive it in a straight line. The terrain is tough on many of my properties, with slopes, irregular shapes.
I locked down the caster spindles, took out the tow hitch pin, and used it in trailer mode for years and years. No, I cannot make true Z turns because of the tow hitch limits. But, I learned to make a two part turn that went quick. Yes, the sulky does make an impact on handling, but it is not a limiting problem. On one or two properties that were flat, it worked pretty well. But, these were too rare to make it useful.
... hope that helps.
P.S. Actually, there was another step in my adventure. Between the one-wheeled Velke, and the BR, I tried a ProSlide. The behavior of the ProSlide and the BR in caster mode was the same -- machine was too light, too small footprint to make it work.