I wrote a Stability Augmentation System once. I forgot what I read that explained it, but I was really surprised how effective it was at stabilizing a Halo Pelican. I ended up not needing a PID for the stick and rudder, since this kept the oscillations under control.

SASExample.cs

// This is a basic dummy version of a Stability Augmentation System. There's a bunch of technical
// terms for this, but I'm not a control systems engineer so feel free to correct me. I found a
// really awesome document (video?) somewhere that explained all this stuff but it was a while ago
// but I forgot where I saw it.

// This is meant to be set up per channel. It's serializable so you can change
// the authority and power of it per channel in the Inspector.

[System.Serializable]
public class SASChannel
{
    [Tooltip("How aggressive the SAS is in stabilization.")]
    public float Power = 10f;

    [Tooltip("How much control authority the AI is allowed. E.g. a value of 1 can command the full range of an input axis in an attempt to stabilize.")]
    [Range(0, 1)] public float Authority = .1f;

    public enum Channel
    {
        Pitch,
        Yaw,
        Roll,
    }

    public float GetDeflection(Vector3 localAngularVelocity, float stickDeflection, Channel channel)
    {
        var sasIntent = channel switch
        {
            Channel.Pitch => localAngularVelocity.x,
            Channel.Yaw => localAngularVelocity.y,
            Channel.Roll => localAngularVelocity.z,
            _ => 0f,
        };
        sasIntent *= Power;

        // Washout filter so that the pilot can override the SAS at high stick deflections.
        // Otherwise, at full stick deflection, the SAS will fight back to prevent turns.
        var authority = Authority * (1f - Mathf.Abs(stickDeflection));

        sasIntent = Mathf.Clamp(sasIntent, -authority, authority);
        return sasIntent;
    }
}

// Here's an example of a very, very basic class using the three channels of SAS. Assume that the
// the pitch/yaw/roll variables are being set by some controller somewhere, and operate on a range
// of -1, to 1.
public class YourCoolAirplaceSpaceship : MonoBehaviour
{
    public SASChannel SASChannelPitch = new SASChannel();
    public SASChannel SASChannelRoll = new SASChannel();
    public SASChannel SASChannelYaw = new SASChannel();

    public float pitch = 0f;
    public float yaw = 0f;
    public float roll = 0f;

    private void Update()
    {
        // The local angular velocity allows you to tell how quickly you're pitching/yawing/rolling
        // relative to your own airframe.
        var localAngularVelocity = transform.InverseTransformDirection(Rigidbody.angularVelocity);

        // The SAS will give its own commands to the axis it's told to. Adding its command to your
        // commands will cause it to have enough control over the stick to keep the craft stable.
        // Remember to clamp to prevent inputs greater than a normalized value!
        var sasPitchDeflection = SASChannelPitch.GetDeflection(localAngularVelocity, pitch, SASChannel.Channel.Pitch);
        pitch -= sasPitchDeflection;
        pitch = Mathf.Clamp(pitch, -1f, 1f);

        var sasRollDeflection = SASChannelRoll.GetDeflection(localAngularVelocity, roll, SASChannel.Channel.Roll);
        roll += sasRollDeflection;
        roll = Mathf.Clamp(roll, -1f, 1f);

        var sasYawDeflection = SASChannelYaw.GetDeflection(localAngularVelocity, yaw, SASChannel.Channel.Yaw);
        yaw -= sasYawDeflection;
        yaw = Mathf.Clamp(yaw, -1f, 1f);
    }
}