Controllers pair with one system at a time. A pad last used on a phone or PC will keep trying to reconnect there, which looks like a broken controller when you switch back to the console.
Disconnections mid-game are usually power or interference. A worn rechargeable pack can report charge and then collapse under load, and a console hidden behind a television blocks the wireless path.
Stick drift is mechanical wear. Dust and worn contacts inside the stick module make it report movement at rest, and while cleaning can help temporarily, the module eventually needs replacing.
A wired connection is the definitive test: if the problems disappear with a cable, the fault is wireless or power, not the controller's inputs.