"Camera movement" in 2D monogame.

I see others recommending interpolation, but that approach does not give smooth results especially after sudden changes in speed or direction.

I'd recommend implementing some simple camera physics to simulate a critically damped spring damper system. Sounds complicated but it's only a few lines of code:

// This code goes in your Camera class
// keep track of both the 2D position and velocity of the camera
Vector2 camera_position;
Vector2 camera_velocity.
// define c, a constant value that controls how quickly the camera follows the target.
// Start with c = 2.0 and try adjusting it up to get a faster camera or down to get a slower camera.
// c must be greater than zero.
float c = 2.0f;

// This code goes in your Camera.Update() method
// each frame calculate the acceleration of the camera.  This will be the sum of two terms:
// the spring acceleration which scales with distance from the target
Vector2 a_spring = (c * c / 4) * (target_position - camera_position);
// the damping acceleration which scales opposite the velocity of the camera
Vector2 a_damping = -1 * c * camera_velocity;
Vector2 camera_acceleration = a_spring + a_damping;
// apply acceleration to velocity
camera_velocity += camera_acceleration * dt;
// round velocity to zero if both velocity and acceleration are near zero
if (camera_velocity.DistanceSquared() < 1 && camera_acceleration.DistanceSquared() < 1)
    camera_velocity = Vector2.Zero;
// apply velocity to position
camera_position += camera_velocity * dt;

Are you thinking of something like in Ori and the Will of Wisps where the camera sort of follows the character but not exactly? https://youtu.be/rkgVb27Iwd8?t=1862 There's some smooth interpolation in the movement.

Or did you have something else in mind?

I've whipped together a simple implementation for you.
Have a look and ask if something is unclear 😊
https://github.com/xnafan/2DChaseCameraSample

How about storing your camera’s viewport size (your desired screen resolution) as a Rectangle, then modifying its .Offset in the Update loop and lerping it towards the player position?