Putting aside the price advantages by switching carriers (there typically are huge price breaks for new customers not available for existing customers), when you come to buy a new phone, there is some price shopping you can and should do that often will produce significant savings. Here is a good general approach based on the Samsung Galaxy J3 (which, unless you need a higher quality and resolution camera has become our phone of choice because of some of its advanced technology and its removable battery (pictured below)) compatible with the AT&T system:
First, check with the carrier's online or store price. It will cost you about $210 and you can pay it at the rate of $7/month.
Second, check with Amazon or other online sellers to find an unlocked version. It will cost you about $125.
Third, check with BestBuy. It will cost you about $150 on a 30-month plan, $120 on a 24-month plan, or $85 to buy outright.
Fourth, call AT&T, ask for the Termination Department. Do not threaten anything, but take the approach that you are exploring the above alternatives as well as switching to T-Mobile which will give you the J3 for free (which it will), and see what deal they give you. Depending on the rep you get and who knows what other factors come into their consideration, you might get a fabulous deal (even one that beats the BestBuy deal) or no deal at all, and always be sure to ask for a waiver of the activation fee and the shipping charge, and be sure you're clear on any re-stocking fee.
Link to Specs, Reviews Etc on BestBuy Website