Coolant system temperature sensor(s)

Background:
(1). No coolant leaks
(2). Engine not overheating
(3). Gas mileage not really changed
(4). Engine running fine

Symptoms:
(1). Coolant fan running almost constantly and goes on and off at unusual times.
(2). At the same time that the coolant fan issue started the temperature gauge started acting goofy! Even when the engine is cold at start up the temperature gauge (even before getting warmed up) the gauge will move to high temperature indication and then slow back down to normal and below.
(3). It will also do this sporadically even when engine is hot after traveling and then settle down!
(4). All the while the fan is running most of the time!

Thank you!

Jimmy stewart