how do i change my airport from bridge mode to nat and dhcp and not have the error of double nat?

You cannot do what you are asking.. the reason it has double NAT is that you have plugged it into a router..

When you use NAT and DHCP you put TC into router mode.. and clearly it says NAT .. so you are back to double NAT.

What you need to avoid double nat is a modem that is bridged or pure type.. ie if you have cable service replace a gateway (or bridge it if allowed) with a modem only device.

Tell us what is your broadband type and modem make and model??

An alternative to bridge and still not getting stuck with double NAT is to use Static IP setup.

See Re: airport time capsule keeps disconnecting

I find this very successful.

Another question is why?? There is nothing wrong with using the airport in bridge so what are you trying to achieve??

You have two simple basic options to avoid a Double NAT error on the network. although a third more complicated option is possible.

1) Configure the "modem" (it is not really a simple modem...it is a modem/router, also known as a gateway) in Bridge Mode and setup the AirPort as a router to do DHCP and NAT. Whether or not it is even possible to set up your "modem" in pure bridge mode is probably a question for your Internet Servicer Provider (ISP), who likely provided the "modem" to you.

2) Let the "modem" (modem/router) do DHCP and NAT and configure the AirPort in Bridge Mode. This is the setup that most users should use. The only real downside to doing it this way is that the Guest Network feature of the AirPort will not work correctly in Bridge Mode.

A third, and quite a bit more complicated option would be set up the AirPort to do DHCP Only.