Edge of 4G and 3G coverage

Getting a new phone and need to know which one I can turn off 4g and use 3g signal. Getting out of Verizon and into AT&T. Was looking at the G3 but it looks like I am on the coverage edge being out in a rural area. I don't want the 4G searching all the time looking and draining battery or using all the data if the house wifi is lesser quality than 4g signal. Is there a better phone without rooting or what ever you call it ? Tech unsavvy here, just an old guy with a computer.

Remember you can turn cell data off if you want. I rememer in the settings it asks if you want to use lte.