engine is smoking not overheating says low coolant but reservoir is full

Hello, the smoke you are seeing could very well be from some kind of fluid or oil dripping or leaking onto the exhaust manifold. This will create active smoke and a burning smell. Only by physical inspection can you tell what the fluid is that is getting onto the hot exhaust and where it is coming from. As far as "low coolant" -- the sensor in the reservoir that indicates coolant level may be malfunctioning. If there is coolant in the reservoir at the proper level -- then yes, it is likely not reading right. The sensor may be part of the reservoir itself. This should be double checked by a skilled technician before proceeding with repairs.