You don't yet know how accurate your meter is. With nothing inside the house powered on, your meter must read exactly what the AC transformer voltages is. IOW with nothing on, then your meter reading 116 volts is the same voltage that Edison Tech measured as 120.
Now power on some major appliances. Your entertainment system is near zero power. Turn on an electric oven, electric dryer, air conditioner, clothes iron, refrigerator, washing machine, etc. If a wiring fault exists, you will see voltages at some other receptacles drop. If not, then you have no problem with the meter to breaker box connection.
That GFCI tester will only report major problems. Obviously, it will not report problems that are causing single digit or ten volt drops. Anything the GFCI tester might report is already reported by your meter. But first learn how voltage numbers change between no load and maximum load.
How high are high bills? How many more kilowatts does your electric bill now say you are charged for? And how many were consumed in bills from years earlier (or by neighbors)? Kilowatts is a critical number. How that number (not dollars) changes over the months is important information.