How can I change my public IP address?

Sorry there is no way for you to change it.  While it is considered a "dynamic" IP as opposed to a static IP it rarely if ever changes.  You could purchase a block of true static IP's and cycle around them however.

About Static IP addresses

Static IP Addresses

Probably not worth it just to access a single site.  And alternative might be to use the TOR browser to access that sight since the other end then won't see your IP doing that.  That at least might allow you to contact them to find out why they are blocking you.

I read somewhere that if you go to http://192.168.1.254/ and press "Reset to Factory Default State", there is a chance it could change. Is this true? 

No, it's not true.  Factory reset does just that, i.e., restores the gateway back into the original state you got it.

On vary rare occasions replacing the gateway with another will get you another IP (see below).  But that's bordering on rumor too except for the fact it once happened to me.

What's wrong with just using TOR?  I once tried to register anonymously at a site so they blocked my IP.  So I simply used TOR to get around them.  Worked for years until the lucky accident when I had to get a new gateway and somehow had a different IP.  But as I said don't be encouraged by that event.  It's the rare exception, not the rule.  And I don't know if some event happened by coincidence outside of my location (VRAD wiring change or whatever) at the same time too that might have caused it.

Just use TOR and not worry about it just for one site.

I think my model is the 5031 and it's currently running this software version: 10.5.6.529707-att

I will try resetting my gateway later. Hopefully it works. 

As I said your IP will not change but be aware that if you do a factory reset any changes you made (if any, for example the wifi passwords) will be lost and you will have to redo those changes.

So there is no guaranteed way to change the IP address at all? Should I just buy a new router? 

Sheeh.  All this for a single site that doesn't like your IP.  Must be a pretty high priority site for you if even the TOR browser isn't sufficient to satisfy your needs.

You can put your own router behind the gateway  to pass through the gateway but you cannot eliminate the gateway.  Whether the 2nd router presents its IP to the world or the gateway continues to announce it's own I do not know.  Maybe someone else reading this can answer that.

You're on your own at this point.

Good luck.

Whether the 2nd router presents its IP to the world or the gateway continues to announce it's own I do not know.  Maybe someone else reading this can answer that.

The second router will just be a private network. All traffic coming from that location comes from the public IP.

@_xyzzy_  Ok, last question, do you know when gateways get new firmware updates or when the lease time for the IP expires? This way I can at least try the reset/unplug method during those times. 

There is no "lease time" for an IP.   And there is no way of knowing your firmware has been updated unless...

(a) you periodically check to see if the version info changed, or

(b) someone else happens to mention it if you are reading these forums or DSLReports uverse forum, or

(c) you have some specific configuration and something goes wrong and you don't know why, or

(d) you see your gateway reboot for no reason and then when it reboots you then find the version number changed!

And I think it's been mentioned you can always force a full download of the current firmware with a factory reset.