That shrieking noise may be the sliding gear going too deep into it's companion gear on the rear differential input shaft. The system is very simple - the internal 4x4 driveshaft runs from the rear diff. housing thru all 3 cases up to the front of the front transmission where it is supported by a bearing and seal and your front driveshaft attached to it (inside that tube). On the rear, it has the same small bearing in the rear diff housing to support the shaft on that end. There is the small section of this shaft at the rear that is splined with the splined, flat gear sliding back & forth - this gear has the groove in it that the shift fork engages and slides the gear back & forth. There are no bushings or synchronizers, the detent for the shift fork is in the plate that you remove from the side of the trans. Nothing on the shaft but the gear and a snap ring to let the gear slide off the shaft for replacement (I've replaced mine once). That rear bearing had gone out on mine and I totally lost the engagement of those two gears and had a lot of noise. When I did the rear split for repair I found the bearing shot and the gear only had a few marks on it but it was so much trouble to get to I decided to replace the gear as well while in there. I did not replace the gear on the diff input shaft.
A rear split is extremely tedious on this tractor, I've been a pro auto mechanic for over 45 years and had a tough time getting the halves back together by myself here, there are several shafts to align while rolling this heavy rear section back together and you have to use silicone to seal the case and keep that sealed well or you'll be losing hydraulic fluid. I finally figured out that because my leg/hip would rest against the 4x4 shifter while turned in the seat working the 3 pt all the time is what put the stress on that shaft/bearing. While doing the major repair I took the shifter off the tractor and used a torch and bent the whole thing over towards the inside of the fender and down slightly so my leg would not rest against it anymore - that was 8 years and 1300 hours ago and I've only replaced 1 shift fork since then from wear & tear. Hope I never have to do that rear split again ...