If you are having IP addresses assigned using DHCP and one of the devices is disconnected from the network for a long enough period of time then the router will make that IP address available to another device to use. If the first device then comes back on the network with the same IP address it had before then you'll get the dual IP address situation. This usually doesn't happen since the returning device is SUPPOSED to ask for a new IP address when it re-joins the network. In your case this may not be happening.
What I do is use static IP addresses for all the devices on my home network. In your case I think this will fix your problem.
Set up your Airport to 'reserve' specific IP addresses for the devices in your local network via their MAC addresses.
Open the airport utility, select your router, click EDIT, select NETWORK.
In the area marked DHCP Reservations: click on the + sign
Type a description of the device.
Use the pull-down menu to select "Reserve Address By: MAC address"
enter the MAC address in the appropriate line.
enter in the unique IP address for each device that you want to reserve. Make sure its in the same subnet (ie. 192.168.xx.yy)
click SAVE
Repeat the above for all your devices.
Update your airport settings.
When you're done turn off all the wireless devices for 5 minutes and then reconnect. Each should then reconnect with the same IP address each time. You can still leave the devices themselves set to use DHCP so that when you take them outsider your network, they'll request an IP address from the new network.
I hope this made sense. If not, post back and I'll calrify anything you need.