The gyro should have an adjustment to limit servo travel... set the gyro to normal mode, move your rudder stick left/right, and adjust the travel limiter to get max throw while keeping your linkage from binding. If you need to recenter the total throw, do it mechanically. Once you've got the gyro setup to the max throw your linkage will allow, the gyro will take care of the rest in HH mode... there should be no drifting left or right, and no trim required.

The Futaba GY401 HH Gyro must be powered on in HH mode, with the heli perfectly stationary, for the HH mode to work at all... I'm guessing it's the same with your JR... read the directions and find out. If you power it up in rate mode, that's all you get... HH mode won't work...