Why desolder from the board if you're cutting the wires down?
As a rule of thumb, I never have the soldering iron on the PCB for more than 2 seconds. Keeping the soldering iron on any longer may damage the board in a number of ways, mostly being cosmetic. If you find yourself absolutely having to hold it on longer to melt solder ask yourself this;
- how long has the solder iron been heating up? My SI is low watts, I give it a good minute between each solder joint to make sure it's at maximum heat. It is not an expensive SI, but that doesn't mean anything to me, it works and I know how to use it well.
- What kind of solder are you using? Flux core is not necessary if you are using soldering flux before each solder joint. If you are not sure what solder flux is, stop now and do a little reading. It's a life changer if you're a beginner! Also, as DirtyBullets suggested, how much lead is in your solder? I like around 60ish/40ish blend at .050" (1.27mm) dia. Smaller is fine for these projects, larger is not. More lead = lower melt point.
Keep asking questions. You're smart enough to find out two buttons were not working. Good work, keep it up!