Odd question: what does trailer sway feel like at first?

Sway can start without notice. A blown tire or tire defect can cause it out of the blue.
The trailer will start "wag" before you feel it in the TV. Then the TV will do the same. You may also feel the vehicle move around in the lane for no apparent reason. Speed will induce sway.
A properly equiped and set up trailer can sway at any time when it never did it before.

I've towed a lot of different trailers since 1968. Most of them just on the ball. No WD hitch, no sway control device. Many of them at speeds that the folks here on RV net would claim I am a reckless danger to society.
My experience is, a properly designed trailer, correctly loaded with adequate tongue weight, hitched in a level attitude, on a tow vehicle that is a good match to the trailer, will tow correctly with no sway.

In other words, get the fundamentals correct first, then add things like sway control and weight distribution hitches to fine tune the entire set up. IMO.

On our old trailer the first time we took it out it was a “white knuckle all the way” trip. On our return we immediately upgraded the hitch to a ProPride which solved the problem.
Question: Is having to “white knuckle” the steering wheel always going to be an indicator of a potential sway problem?

Dan, do you have wide tow mirrors on your Tacoma ? If so, you should be able to sight down the sides of the trailer.

Next time you re towing, at highway speed, hold the steering wheel totally straight, and sight down the side of the trailer. You may see it very slightly "moving" side to side. Note I did not say swaying. We'll call swaying an uncontrolled and increasing amount of side to side movement. Which is of course bad.

By comparison, a slight amount of movement is normal. Take note of the movement of the trailer sometime on a calm day after a car hauler type 18 wheeler has passed you, and is out front of you 200 to 400 feet. The air off them is very turbulent, and you will see your trailer doing a little side to side dance.

Sway is like a "worse version" ( greater amplitude, lower frequency at first ) of this dance, that gets progressively wider swings.

Train your eye to see this side to side dance at a glance in the mirror. Understand what it looks like, and think it thru, that if it's getting worse, you are heading into a potential problem.
If it does begin to sway, steady throttle at first, coupled with a "light application" of trailer brakes manually at the controller, followed by easing off the gas will bring it back straight.

That's my experience.

Usually when a truck passes the front of the tow vehicle will turn toward the truck (all by itself) and the driver has to correct to the right to avoid a collision.
This also depends on which way the wind is blowing...

It'll feel like the back end of your vehicle is slightly floating and moving side to side .

When you get passed by a big truck, that is a PUSH feeling. It is not sway. Sway is an oscillation.

I've only come close once, with a combination of going a little too fast, an odd road surface, and full tanks, which probably unloaded the tongue a little bit. First thing I noticed was the steering feeling wonky, and at the same time, could see the trailer appearing and disappearing in the mirror on the driver's side. I immediately let up on the gas and it stopped, even before I had a chance to use the brake controller.

As it happened to me, i noticed the trailer swaying in my mirrors , then i felt that force pushing my hitch which made the steering feel lighter.
All three forces seem to occur simultaneously. As I let off the gas, the trailer fell back in sync. It was a small trailer at about 2000 lbs with no trailer brakes . If I had brakes on the trailer I would have let off the gas and applied brake pressure to the trailer brakes. Its usually NOT a good idea to speed up or use the brakes on the tow vehicle .

If you have not had any sway yet, then your trailer is in balance or loaded correctly. Therefore (besides a flat tire or other mechanical occurrence) strong crosswinds or a bow wave from an overtaking truck or bus would probably be the reason why your trailer would sway.