This is repeating ether’s point but firstly, in pushing pand, each player are constantly trying to find the opponent’s weekness, that is the part of body which is not aligned and being relaxed properly. Now if you, yourself, are not relaxed and aligned, how do you expect to find your opponent weakness.
But in regard to sensitivity, in taijiquan, sensitivity is not about you having the hypersensitivity of Daredevil, rather it is to do with you being song (relaxed).
In taijiquan, being sensitive means you can follow your opponent. The ability to follow your opponent is directly related to your ability to be song. Moreover, if your arm is not relaxed, you tend to constantly push part of your arm into opponent’s arm or body as if it is a blade, which send signal to your opponent what your intent are. So in taijiquan sence, you are less sensitive than your opponent.
I had a chat with a judoka who spar with very high grade judo players. He said that when he got into holding position with these highgrader, he hardly felt their arm flexing their muscle. But whenever he tries to do any throw or push, it was quickly blocked. Then suddenly the highgrader snap their wrists and bang, he was on the mat.
I also read somewhere about ancient Greek’s Spartan quote. It was about this wresling instructor. He had one particular student who kept his wrist very rigid so instructor gave very nasty wrist lock. The kid couldn’t use his wrist for few weeks but the lesson was well learned.
So when you practice sensitivity, concentrate on song (relaxation). You also mentioned that when you push the guy, you get easily pulled in. This is to do with you being overextended, that is when you push, your back leg is not relaxed so you tip your self over rather than sinking into your front leg.
What is great about pushing hand is that when you make particiular mistake repeatedly in pushing hand, that mistake can also be seen in your form performance.
I would imagine that when you perform your form, your back leg is not realxed and possibly shoulder is tensed up, arm held too high and elbow are not sunk. So you need to pay particular attention to that part of your body being song.
So practcing form let you fix your fault in your pushing hand and doing pushing hand improve your form performance. Hope this help.