Xbox Live and WiFi

I have pretty decent speed internet, and have been with Cox for about a year. Recently, in the past two months my Xbox Live has been sporadically stopped connecting to certain things. I can no longer join or start parties, and lately cannot even join most friend's games. I have done all that Microsoft has suggested I try, and my NAT type still reads strict and I face continued connectivity problems. My internet connects fine to all my other devices. I am not well versed in computer/Internet troubleshooting, but a friend suggested that my connectivity issues may stem from my internet. Are there any fixes so that I can begin to play on Xbox Live again?