A lot of it is going to be game-specific, and spending time tweaking the control settings until you find what feels responsive to you.

The rest of it is going to be technique, and a lot of trial and error to find out what works best for your play style. For instance, I can't do fast-paced, twitchy movements on a controller (even things that are technically possible to do on a controller; I just don't have the dexterity anymore), so I have to adopt a different play style when using a controller. I usually will go for a more support-based role, if possible; opting for long-range weapons/abilities, and playing a more patient, campy game. I play slower and more methodically this way, and try to position myself so that I don't ever get into the situations where I need to react to somebody closing the gap on me in the first place.

For me, it's an entire mindset shift. If I play the same game on M/K, I'll be playing with a much faster, reaction-centric style instead of one where my movements are more premeditated.

Some other tips will be learning to do things like using your left stick for fine-tuning your aim (you can get very precise horizontal micro-adjustments by leveraging your player's position, which can be useful for getting your shot off before the other guy does), experimenting with gyro controls if that's an option for you, or trying joystick extenders (small gadgets that clip onto your sticks to extend their effective length, which may make aiming easier).

As far as what to practice in, I don't know of any aim trainers that are designed for controller, so I'd say you should just practice with a game that you either don't care about or where it doesn't matter if you lose a bunch. I'd recommend The Finals; it's free to play, the default quickplay mode is active and puts you into a match quickly, and it's super low-stakes so you don't have to feel bad about experimenting during a live match. Your teammates don't have loot drops or anything hinging on your success, so if you play badly, nobody cares. And it's got pretty robust customization options for the controller settings (dead zones, acceleration curves, etc), which can help you figure out what settings you respond best to and what to look out for in the settings of other games. It has a huge variety in movement/weapon options, so you'll end up developing skills/habits that will transfer over to other games quite easily.

I didn't mean to weirdly steer this into becoming an ad for The Finals. But it's a very controller-friendly FPS that I think will be beneficial to practice with. I think it's also pretty fun, but that's subjective.