A couple of things as former T-Mobile Tech support (over 5 years now, so…)
One. While most phones aren’t going to have an issue, you need to make sure they’re unlocked, especially before you switch, which is normally an issue if you purchased through your current carrier. It’s not normally hard to do, but if you do it after you drop your current carrier, it can be a pain.
Two. Most phones (again) will work fine on a different carrier, but you want to check your specific frequencies supported vs the ones that T-Mobile supports in your area. T-Mobile is making a big push in its mid-band 5g tech for their coverage, which means older phones may have a problem giving good coverage throughout an area. (this is the sort of thing that @kenobi_65’s mom may ben having an issue with, or just a dead zone in coverage)
Three, AT&T’s price is really on the high end. You can and should absolutely shop around, most unlimited coverage plans run around $45 (or less) per line per month, often less with things like autopay/employer/AAA/etc discounts. So even if you don’t do T-Mobile, you can certainly do better.
Okay, now, onto recommendations. If you don’t need unlimited data, you can absolutely do better if price is the focus, but let me put out two T-Mobile Options for your review, but if there isn’t a promotional price issue with the AAA, it is quite competitive.
Would be the T-Mobile’s Essential package, which, I would bet, is similar to the unlimited plan they’re offering you. It’s 45 per month per line with autopay (generally $5 per month more otherwise), and you can be subject to de-prioritization if you use over 50 Gig of data in a month. It’s not exactly throttling, but it can be an issue for some users. Still if you’re using 4G of data month, you’re extremely unlikely to ever hit that.
Alternately, you could go with the T-Mobile prepaid
https://prepaid.t-mobile.com/prepaid-plans
Where 2 lines with 10G each of data runs you $70 per month, and you still have more than you have with AT&T for far less in cost. But the prepaid plans often leave you with a lot fewer bennies. The biggest being left out of nearly all promotions for things like phone upgrades/trade-ins, which normally only apply to the higher end plans. But if your upgrade needs are few, no biggie.
Again, I -bet- the AAA plan is pretty much a discounted version of the essentials plan I linked first, and it’s a very good price if you clear the first couple of issues I linked above.