If you look at the T-handle in its free state, the button is depressed flush with the bottom of the T-handle. This is incorrect and why your shifter rod will not pre-load the detented rooster comb comprising your shift gate. You may be able to take the T-handle apart and see if the internal spring has been dislodged or is broken. I sold a T-handle to a fellow E-32 owner who had this same thing happen as I upgraded to an E-38 pistol grip. The weight of the rod is insufficient to overcome the internal friction of the button mechanism. Driving the car without positive engagement is downright dangerous and could be disasterous for your trans if you were to inadvertently bump the shifter and select Reverse or Park while driving without any lock out.
Try taking the handle apart...if unsuccessful, replace it. Perhaps Rigmaster has one from a parted E-34.
HTH.