The problem is so common among modern bikes that I've seen videos where experts say you need to practice taking one hand off the bar on a deserted road so that if you need to take your hand off the bar to signal or for some other reason, you wouldn't crash.
It was so severe with my Radrunner 1 that until I modified it, I had tennis elbow from trying to control it when I had to take one hand off the bar to signal. The biggest problem was the seat position. The pavement around here is rough, so I needed to stabilize my upper body. With only one hand on the bars, it took tremendous torsion on my elbow to brace myself while keeping the bar straight. It wasn't safe.
Moving the seat back several inches fixed that. Now I could brace with my feet against the pedals, making control with one hand easy and safer. I could even ride with no hands, but I didn't like it. There were other problems.
When a bike tips, the contact patch tire of the tire moves to that side of the tire's centerline, and that pulls the bars to that side. With no hands, that could cause a crash if it pulled the handlebars too far. Trail counteracts that tendency. My Radrunner has 6 cm of trail. That would be fine for a tire an inch wide, the Radrunner tires are 3.3 inches wide. I think more trail would make it safer.
Then there's head angle, the angle of the steering axis from horizontal. The farther it is from 90 degrees, the more turning the handlebars moves the contact patch in that direction, with respect to the rider more than with respect to the tire's centerline. Leaning with no hands can cause the contact patch to move just far enough to bring the bike vertical and return the steering to it's original direction. It's a sort of autopilot.
My Radrunner 1 has a 70 degree steering head angle. There's not much autopilot effect, and if it started to turn sharply I'd need to grab the bars fast, not only to steer but to stabilize myself. For some reason, steering heads these days are often even steeper.
I grew up with English bikes having a 66-degree head angle. Turning required a bit of countersteering to get the contact patches out from under me so I could lean. On the way back from the municipal pool I'd come down a side street with a grade of about 20%. On a bicycle, that seemed a bit like freefall. The stop sign at the bottom required heavy use of the brakes. They were rim calipers, so cooling was excellent. It would have been dangerous with a coaster brake.
I used to spare my brakes by grabbing two corners of my towel with each hand and holding it behind my head like the parachute on a dragster. When it opened at that speed, it would jerk me. The bike would continue straight until I provided handlebar input. It was a far more effective air brake than initially expected. I needed only a little use of the calipers as I approached the stop sign.
I think head angle is the most import factor in hands-free riding, but changing forks would not have changed your head angle. Your front tire may look well inflated, but I think its pressure is lower than when you used to ride hands free. With less pressure, the contact patch on that fat tire would move farther from side to side as the bike tipped, and that would pull the handlebars in that direction. As you say, squirrely.