Sempiternity
Smash Lord
You're in luck. I was just taking apart my controllers for some cleaning and still have the parts exposed. Installing a new stick is dirt easy. Sorry for the piss poor quality; yay camera phone and weird lighting.
New sticks will come in a small grey/black box thing about the size of a ping pong ball or two. They're made this way to be really easy to replace and fiddle with.
The thing in my hand is the circuit board within the whole stick module, so your new stick will be that grey/black box, but with wires coming out, ending in the blue thing (might be a different color for you).
All you gotta do is dig your fingernail or something in the little groove, and slide the connectors apart. Simple. It does take a little force.
Just another shot showing how they fit together.
As for your button sticking problem, just take the buttons out, and clean the edges of them, as well as the edges of the frame containing the button. Rest assured you'll find tons of sludge and grime in there; a fine collection of all the dirt and food that's been on your hands.
New sticks will come in a small grey/black box thing about the size of a ping pong ball or two. They're made this way to be really easy to replace and fiddle with.

The thing in my hand is the circuit board within the whole stick module, so your new stick will be that grey/black box, but with wires coming out, ending in the blue thing (might be a different color for you).

All you gotta do is dig your fingernail or something in the little groove, and slide the connectors apart. Simple. It does take a little force.

Just another shot showing how they fit together.
As for your button sticking problem, just take the buttons out, and clean the edges of them, as well as the edges of the frame containing the button. Rest assured you'll find tons of sludge and grime in there; a fine collection of all the dirt and food that's been on your hands.