Motor is whining but does not agitate

I'm not sure if you can buy parts for the transmission or if you'd even be able to repair it to working order anyway. Being that it's a $300 assembly, isn't it new washer time? If you can't find individual parts for the transmission, the point is moot. I had one with a bad main bearing; it would agitate but couldn't spin under load. I'll be the first to admit that I'm just a parts swapper; entirely disassembling a washing machine would not make me any money and would probably optionally cost you the same, in lost time (and more parts when it breaks again next year; you'd still have an old washer).

It sounds like your washer needs its main bearings replaced. My top load Sears Kenmore HE washer had the bearing seals fail and allowed water to get into the bearings. Over the course of a few months the washer, in high speed cycles, sounded like a freight train passing by in the next room. It was LOUD just like you are describing.

A sure fire test to check if it is the bearings is to rotate the washer tub / agitator by hand, when the washer is OFF, for a few turns. If the rotation has a grinding or roughness feeling then it is almost for sure the bearings.

I took on the repair myself and purchased the replacement parts on line for ~50$US. It also took a special tool to press in the new roller bearings. I found out that the bearings were industry standard size roller bearings (my washer used two bearings each of which was a different size) so I went to a site that sells a bit higher quality bearings with known names than the brandless import types used in the original washer. The special tool was a bit expensive at ~85$US but made the install very easy.

My suggestion is to get your washer model number and do some Google searches. When I did that I even found that the results turned up several U-Tube videos showing the steps in the bearing removal and installation.

After the new bearings were in place washer runs better than a new one. The thing is now so quiet I'm never sure just when the last load of laundry is completed.