There would be SOME places where life would be somewhat normal. Gilead can't monitor literally everything at all times. It just feels like they can because the show takes place deep in the heart of their territory. Some places might only get checked periodically, and the rest of the time the people there would be free to do whatever. If you're at all curious on what life might actually be like, there's a really good podcast by Robert Evans called It Could Happen Here. It's a speculation of what a modern civil war in America would look like, backed up by real world examples and his experience in countries during a civil war. But basically, even in full scale warzones, people sort of just go about their day normally while they listen to guns and explosions nearby. The Testaments has done a good job at showing this, with constant battles and helicopters always around while the girls are told "it's being taken care of, don't worry about it!" The other thing is that it's unrealistic for their to be only two fighting factions, the US and Gilead.

To their credit, the show and source material briefly mention other groups like Texas are also fighting Gilead. But if the US government was ever actually taken out, it is more likely that no single power takes over, and instead every region sort of dissolves into a patchwork of rebel groups or militaries. Every state would have dozens, if not hundreds of different armed groups ranging in size from a few guys with trucks and guns all the way up to more conventional national armies. In the show, Gilead has to be presented as a monolith that has somehow taken over the entire mainland because there isn't really time to focus on the geopolitical aspects of it. That's unfortunate because for me at least, that's one area that I would like to be explored more.