Welcome to the Forum. We're here to help and will welcome your help as well!
I looked 27DS up in Jayco's Brochure Archive and wheelbase is 182" with coach length about 28-ft. Wheelbase is 54% of length. Could be better, but as a statistic it's adequate for tracking on the road. But, what is your Weight Distribution? Have you, or can you, weigh this coach, set up and loaded as for your trip? Once you do, look up a load/pressure chart like
this from Michelin (brand doesn't matter as long as Size and Load Range match) then adjust tire pressure to actual weight. If your front tires are overinflated for weight carried, it's ready to wander if it already isn't.
You say "Sway" and I'm sure it's hard to tell, but... Sway as in Rocking side to side, or more of a Weaving motion on the path of travel? They're related. A weaver will sway. One that sways (basically from the weight of the heavy RV body in the rear) will cause the steering geometry in the front to change, particularly on a Ford chassis. Then it'll weave...
In summary:
Your 27DS has about the same rear overhang as our Escapade 31A, but the extra length is the 31A shows up in 36" more wheelbase. Longer wheelbases track better in their own right. In an RV they also shift adequate weight to the front axle which helps tracking.
I think you'll come out "front light" when you weigh. You have a slide back there that our coach and other Jayco's do not.
A Rear Track Bar should help. Replacing the Sway Bars with heavier duty ones from Roadmaster or Hellwig will cut sway so the steering geometry won't change so much.
You should check Front End Alignment and try to get at least 5* or more Positive Caster on the Left and 5.5* or more on the Right. Be sure Toe is Zero or slightly IN (Positive) by 1/16" or so.
Do you do your own RV work? Can you get to Hendersons Line-Up in Grants Pass? They're the gurus, and recently told me that a rear track bar and a Safe-T-Plus steering stabilizer usually clears up most Class C handling problems.
And be sure there's no wobble in your towing rig at the Receiver, any Drops, Adapters, etc. as noted in previous post.
I should add, is the front end of the Equinox in decent shape and alignment? Years ago I transferred trucks, and sometimes we towed a Dodge Van. Mine had a solid front end and towed fine. A friend had a worn out one that wandered behind the truck towing it. A wandering toad would be like a loose tow bar.
If God's Your Co-Pilot Move Over, jd
2003 Jayco Escapade 31A on 2002 Ford E450 V10 4R100 218" WB