First thing to check is connect your Mac or PC to the hotel Ethernet jack using an Ethernet cable and make sure that you can get a good, solid Internet connection that way. Be sure to turn off the WiFi on the computer for this check.
If you can, then you can move to the next step below. If you cannot, you will need to get the computer working with the hotel Ethernet connection first......before you can do anything else. Unfortunately, most hotels have gone to wireless, and even if there still might be an Ethernet port in the room, it is often not enabled, or not working correctly.
Things are going to be difficult if your computer does not have an Ethernet port to test the hotel Ethernet connection.....since you won't be able to know whether the hotel Ethernet port is really working correctly or not.
Assuming that the hotel Ethernet port is working correctly with your computer, reset the AirPort Extreme back to factory default settings and start over again with the setup. Apple's setup "wizard" on the Mac or iPhone/iPad will probably configure the AirPort Extreme in Bridge Mode automatically, since it will likely detect the hotel router and try to avoid the Double NAT error.
Let the "wizard" do its work and try the AirPort Extreme that way. If everything works when the AirPort Extreme is set up in Bridge Mode, you might decide to leave well enough alone.
If you want to try to set up a "private" network, you will need to change the Router Mode setting on the AirPort from Bridge Mode (Apple calls this "Off (Bridge Mode)" to DHCP and NAT. This will produce the Double NAT error that you have seen before.
Don't worry about the blinking amber light on the AriPort at this point.....you can correct that later. Power off the AirPort, wait a few seconds, then power up the AirPort again and see if the AirPort will obtain an Internet connection or not. If it will, then you are all set. If it won't after a few tries, you may have to run the AirPort in Bridge Mode. Nature of the beast at some hotels.....they just don't want you running another router on their network.
I doubt that it will help to use the Google DNS servers, but no harm in trying.
