DNS Issue TLS handshake slow performance

Try changing your DNS server to 8.8.8.8, or 4.4.4.4, and if that works your section of ATT'S network has some bad domain name servers . It happens.

Locally, I can tell you that our DNS servers go out with some regularity. I finally moved to a third party and have gotten much better internet performance as the third part DNS servers are much more responsive.

Well, I am certainly not the expert on this. 

However, I did set the DNS locally to google 8.8.8.8. It does not resolve the issue. I saw ATT has this DNS prefix as well, not sure what it does. 

The problem is, for the same website, when I use ATT, I cannot connect to it. The moment I switch to Spectrum, it works like a charm. (yes, I have both services because I was going to switch from Spectrum to ATT).

I have talked with at least 3 tech support supervisors, no one can solve the issue.

I have one technician coming out to my house, all he did is calling the support center and tell me "this is an external website, we cannot support". I asked what is NOT external website. He said you know like Att.com. I mean, come on, almost all the websites are external then. 

So you have two WiFi networks and switch between them? If so, your computer may be keeping some of the first network settings when it moves to the second. What operating system is it? (Try rebooting to see if that fixes the issue.

Also, is the website that you are trying to get to unusual in some way, I.e. is there some reason why AT&T might have blacklisted it?

i had similar issue and changing it too  8.8.8.8 helped me.

speed is improved

Well, it is not blacklisted. Please read the full thread. Because this only happens during certain time of the day. If it were blocked, I would lose complete connection. 

I have multiple Desktop/laptops. Mac Windows etc.

Restarting my own device will not help. Restarting att gateway will not help. 

I tried local DNS setup overwritten method. It worked for a short period of time and then next day. It does not work again. (Not able to connect during certain time of the day even there is no change on my side.)

Also, caching is not the problem. Did a full flush. Not helping. 

Again, the original connection was to ATT and then when it stops working, switching to Spectrum will work, switching it back does not work. 

So no, not the caching issue. 

Do a tracert to one of the problem websites when you can connect, then do one when you can't and post them here.