One thing you can also try is heating up the inside of the shift lever to loosen old grease. I am not saying this is a 100% safe and fool proof solution but I tested it with an older DA shifter and used a lighter and put it towards some old grease and it worked and I just wiped down the shifter afterwards. I tried to be very careful not to hit any plastic and it didn't deform anything and the shifter performed excellently after that. Soaking can work but sometimes just getting the grease and oil flowing again can help.

Again be aware this can be dangerous and can cause issue so be careful. I happened to have a shifter that I thought was dead so whatever I did wasn't a big issue. I also didn't spray it with anything that would add flammability and did it in an area where I could easily put out any small fire should one occur.