A few things come to mind... First of all, how much tongue weight do you have on it? If the camper was empty, it'll likely haul a whole lot different when you load it up and tow it. Being too tongue-light to start with, combined with using a weight distributing hitch to further lessen the load on the tongue can certainly cause that.
Are you running a separate friction sway control, or are you running a WDH with built-in sway control? The latter is FAR, FAR more effective than the former on a travel trailer that size.
You really need to take the vehicle to a scale and see what your weights are. Any truck stop will have one you can use, and it'll cost you about $11. Load the camper and truck up with all your gear, fluids, kids, coolers, etc you would normally take for a weekend. Take it to the CAT scale (that's the certified scales here in the states... I'm sure CA has something equivalent), hit the button, tell them it's a 'personal' vehicle, and get the weight. Pull around and pay the $9 to get your first weight.
Next, release the tension from the weight distributing bars and roll across the scale for your second weight. Tell them it's a re-weight of (last 4 digits of your first ticket) Go back in and pay your $1 and get that ticket.
Finally, drop the trailer in the parking lot and roll back across the scale with just the truck, tell them it's a second re-weigh of (last 4 digits of your first ticket) and pay the other $1 to get those numbers.
Bring all three numbers back and post them here.
IMPORTANT: Make sure that when you stop on the scale, the front axle is on one scale pad, the rear axle is on a second pad, and the trailer axles are both on a third pad. You want to get all three weights.
From your numbers, you can calculate the ACTUAL tongue weight of the trailer, the ACTUAL gross weight of the trailer as loaded for a trip, and the ACTUAL effect the weight distribution bars have on the axle weights of all three axles. It'll also tell you if you're tongue-light, tongue-heavy, or over on either of your tow vehicle axles, all of which can affect the handling.
As for the tires, upgrading to stiffer-side-wall tires or a higher load range can sometimes help, but I wouldn't count on that to prevent or combat the trailer sway.