Hi,
You've trained yourself to associate pitch with certain names (A,Bb etc) and also with an awareness of what it feels like to play them on the piano. You simply need to associate what your hearing with your harmonic knowledge.
Instead of hearing "Cmaj7, Dm7, G7" you hear chord "I, II, V". By doing this, you're intellectualizing only to the same degree as you were previously by naming the pitches. Then, as the previous post says, you adjust your ear to the new key and play "I, II, V" which you will recognize instantly.
Melodically, you need to do the same thing, so be aware of which degree of the scale the melody note may be. If your harmony is good then you'll quickly forget that you could never do this.
A good friend of mine had excellent perfect pitch and had to train himself to think this way......,which I think that we should all do anyway.
Joseph Hofman's father apparently had such good perfect pitch that if a tune was played in a different key, he simply couldn't recognize it! Now that's a pain.
Ps any notion of concentrating on the page and avoiding listening, sends me round the room screaming. You merely adjust the way that you think about WHAT you're hearing.