Hi from JustAnswer!
Wherever the coolant level sensor is located (either on the recovery tank or the radiator) the level can be reported incorrectly if the sensor is covered with deposits or slime from old antifreeze. Sometimes flushing the system won't remove deposits, and cleaning the sensor by hand is necessary to get it to stop reporting low coolant if the system is in fact full.
Additionally, chances are that if deposits have built up on the sensor, an equal amount is built up inside the radiator. Enough deposits will prevent the radiator from cooling at all, leading to consistently high engine temps. If you find that the car runs above 190 degrees even while driving with no thermostat at all, the radiator is probably blocked.
The sensor itself isn't the cause of the problem, certainly. If it is reporting low coolant when you assert the level is full, the short answer is that it is contaminated and must be replaced. If a contaminated sensor is found it follows that the same contamination is elsewhere in the system, possibly the radiator. If the radiator is blocked, with these contaminants, the car will run hot. If contamination is the cause of both conditions then the flush did not accomplish what it was advertising, though a complete flush done properly would have a positive effect on any system that is not seriously blocked.
Confirm that the coolant fan comes on at the correct temperature, cycles itself off after a couple of minutes, and stays off for at least two minutes. A cooling fan that remains on without cycling indicates poor heat transfer in the system, meaning a defective pump, thermostat, blocked radiator, airflow issue or contamination.
Air bubbles from a leaking head gasket can trip a low coolant sensor and also be the cause of running hot, and can happen on the same day. Suggesting that this is the problem without first performing a cooling system performance test (described above) and pressure check is guesswork, but I do recommend a pressure test to eliminate bubbles as a cause. Note that a pressure test simulates (and assumes) the vent pressure cap is also working properly. A 70/30 mix of coolant boils at 230 degrees if not pressurized, and so the pressure cap assembly should also be tested to its rated maximum.
A head gasket leak can send air/exhaust into the system without overheating the engine, but a large enough bubble will interfere with circulation and then the car starts showing signs of overheating - it's like an air embolism. Many times a car shows signs of high system pressure after only five minutes of driving, and even though the temperature is low, the venting out the cap can look like overheat (imagine a 'cold boilover' - it happens). The gasket leak would have been caused by the overheat but the lasting effect is inability to circulate coolant or exchange heat or even keep coolant in the bottle because it is regularly displaced by unwanted gases. In your case it's not likely that the gaskets are damaged but consistently higher than normal temps have to be addressed so you aren't in the danger zone for that kind of damage.
Damaged head gaskets come from hotspots on the cylinder head, which circulate coolant vapor through the system and makes a stange percolating noise when it happens. Shortly after that the system boils over but if the system is reasonably full, damage doesn't happen unless the car runs low enough on coolant to stop circulating. Whether the bubbles are exhaust from a leaking gasket or vaporized coolant, the sensor will detect it as a void and turn on the coolant level lamp.