I have tried two different thermostats and have the same issue. It's set to 77 degrees. The digital display accurately reflects the temperature in the house. But, the AC doesn't kick on when the temperature goes above 77 - it kicks on at 79. And, it keeps running until the thermostat goes below 75. So the temperature is fluctuating by 5 degrees all day and night - hot cold hot cold - instead of staying at 77. I thought a new thermostat would fix the issue, but it's the same. Is this just how thermostats work? Does everyone have this problem?

No thermostat with keep a temp precisely at its setting... or your A/C or furnace would be running every 5 minutes. But there is a upper and lower limit, and while 5 degrees is a bit more than mine varies... I don't think its really all that excessive. On some thermostats you can adjust that range a bit.

5 degrees is way out of line. My Honeywell never goes over 2 degrees and then only on very hot-96- days. Were the thermostats you used the same brands? If so, contact the manufacterers tech line.