Assuming those are SPDT switches, and they sure appear to be, why not just attach one pole of each to +0V and the other to +5V and the common of each to a digital pin on the Arduino. Set the pins as input and poll them. When a switch is set to the +5V pole it should register on the pin as HIGH and when it is set to the +0V pole it should register as LOW. KISS.
Yes, you can attach resistors, but the Arduino isn't going to pull significant power anyway.
SPDT: Switch - Wikipedia
What is going on with the Adafruit demo is different. It is a SPST switch that is being pulled up to +5V through a 10k resistor when the switch is open and is draining that +5V to ground (thus is looks like +0V to the Arduino) when the switch is closed. SPDT is easier to implement, you don't need to pull up or pull down although there may be a middle position on the switch that will connect neither pole to the Arduino and in that position the pin will float and might give either value. But for this, a simple project, I would just connect it as I said, that is how I started out.
One last edit. If you don't have this, you certainly should buy one:
This will tell you if a pin or wire or whatever is set high or low or has no signal. I bet if you checked the Arduino pins and connections along your circuit with a logic probe the circuit / Arduino pins would not be doing what you think they are doing and it would be a huge hint to let you know exactly what is wrong. I suck at soldering. Whenever I solder something I go over it with a logic probe before I start plugging expensive parts into my sockets.