ESP8266 + DHT11 AWS IOT Thing not updating thing shadow - was working before
I was facing a similar problem: trying to update a thing's shadow on AWS IoT with no success. After some research, I found out that I was using an outdated library. In this question they advise to use heskew's aws-sdk-arduino, on iot-get-shadow-and-cleanup branch:

https://github.com/heskew/aws-sdk-arduino/tree/iot-get-shadow-and-cleanup.

Additionally, I think you have a little error on your code. As far as I know, AWSPath, it should look like:

iotClient.setAWSPath("/things/esp8266/shadow");


At least that works for me, and they use it this way on this video.

That's all I can tell... I am learning at this moment as well.
Hope it helps!!