There are a lot of good suggestions here, that you can take advantage of, so I'll come at it from another perspective.

With mouse and keyboard, positioning is a snippet of what we use when playing and is more of a tactical spacing. With controller, it is a necessity. This means that as you are playing first person shooters (or third person with controller), your characters movement will be 75% of what you're actually aiming with.

On a mouse and keyboard, if you're slightly off center with a sniper, it's a simple adjustment to move to the left. Move 1cm.

On a controller, if you're slightly off center, suddenly it isn't as simple, because the joystick is overly sensitive and so to move 1cm is a lighting fast action input, meaning that you're almost guaranteed to overshoot it, unless your joystick sensitivity is super low. Or, on the opposite end of it, if you try and move the control stick very gently (more on this later), it's not necessarily a consistent input. This is where aim assist would come in, as aiming down your sights would center it on the enemy, but I think it's a bunch of bullshit and so we'll ignore that. Instead of moving the joystick a micron of a second to properly position yourself, moving your characters body (WASD/left analog) is almost always much slower and fine tuned.

What this means is that as you're playing games, instead of holding W and maneuvering with A, S, D for counter balance or strafing or whatever, the joystick instead is 60% of the time holding forward, 20% of the time slowly moving in a direction to position yourself better for aiming, and 10% staying still (letting go).

Another element here is the concept of analog itself. When you're holding W, it's always 100%. When you push forward, (game depending) it ramps up from 0% towards 100%, which means that if you turn left or right, chances are that your character might slow down too, because you may be pulling down as you move. What you can take advantage of here is utilizing slow movement to always keep your character moving, which will help prevent being hit and will get you more used to fine-tuning your aim through your movements.

When I play games on controller, I always try and use gyro, I always keep the gameplay focused on the movement first and foremost, and the analog stick at that point almost purely becomes a look/view stick over a "this is my main form of getting headshots", where your look inputs are based on getting into the center of the general area you want to aim at as quickly as possible, while letting the gyro and the characters body finish it off.

Finally -- PLAY. Not the game, PLAY with it. Feeling weird? Move your character in circles while bunny hopping to get the feeling of the mechanics for the game, then be silly with the aiming and wiggle the joystick around to familiarize yourself with aiming with the movement wobble. Whether it's Max Payne, Smash Bros, Doom, Vanquish, Fortnite, all of these games can be manipulated by playing with the weird quirks of their engine.

Finally finally -- I also have a harder time with FPS games on the Steam Deck compared to other methods. Doom 2016 on my Switch was fine to get used to, but on the Steam Deck some did feel odd about it. I don't have the other modern consoles and their joysticks aren't super familiar to me, but I think it may be that the Steam Deck's analog sticks feel like they have a larger travel distance (particularly compared to the Switch of course). Something you might consider trying is the Flick Stick input for the Trackpads, although I personally really, really enjoy low-friction trackball mouse input. Swipe+Tap to aim is just so good and being able to move the view, let go and have it keep moving based on the intertia I input is just perfect.