Pay attention to how the switches align in front. Now, with a pencil iron, set to about 750 degrees, and the solder side up, put some pressure with your finger on the switch from the other side to provide force so you can "tip" the switch as the solder releases. Now, apply soldering iron to alternating pins of the switch from solder side. Get the solder to "wet" with pressure then go to other pin.. One side of the switch at a time, not all 4!!! The first one took 5 minutes... By the fourth it took about a minute.
Now, clean the solder holes. First use "solder sucker" to pull out the solder or use pencil iron to move the solder away from hole. Running a steel wire... A toothpick....whatever will help pull the solder clear... Don't ream out the holes as they will help the new ones stay in place as you solder.
Properly align the new switches.. Press into hole until switch bottoms on the circuit board. If it is resistant, verify you align switch with the pattern traced on the board. Don't flex the board.
Once all are bottomed on the board, flip over and solder in place.. A hot iron like mentioned here will solder quicker.. A tip.. Have a small drop of solder on the iron and touch the pin and then when it starts to "wick", drag solder to the board.
I usually let it cool if having trouble with the solder.. Localized heat is what is preferred and a good, hot iron with good heat transfer (solder ball on iron tip), you will not melt the wrong thing.
Peace and Blessings..