Double NAT error on new AirPort Express
You're connecting a router to a router, so you will get dual NAT.

The shared router is translating the single IP from the ISP into a shared network for you all to use. If you connect a second router to this to connect multiple devices, then you are creating a second network, and network address translation (NAT) is happening twice.

I'm not sire if the airport express supports bridged mode, but you could try this to see if it will allow you to connect multiple devices over Wi-Fi with the main shared router distributing the IP addresses. If you can do this, it should solve your dual NAT problem.

Dual NAT shouldn't however cause intermittent connection, that's more likely interference from other nearby Wi-Fi devices, so I'd suggest selecting different channels (if possible) until you find one that gives you a more stable connection.