All the above should get it working for a while. You will probably need to repeat periodically as once the cleaner/lube/solvent/penetrant/displacer evaporates, that cursed Shimano OEM grease will gradually re-solidify and re-gunk.

If I'm sending a bike out for sale or donation, I prefer to try to get all the old grease out, which requires removing the shifter from the bike, an ultrasonic cleaner, patience, and maybe a freezer (optional). Details for the "more thorough" method are at De-Gunking Balky Shifters - For the Long Haul