How can I test whether my furnace issue is the limit switch or the thermostat?
If the high limit switch was bad, the furnace should not fire. When the high limit switch trips, the flame/heating element will be shut off and the blower will continue to run to clear the heat. The blower will continue to run until either the system is reset, or the limit switch closes.

I believe high limit switches are normally closed, and they should always fail open. Which means when they go bad, or the temperature gets too high they open the circuit. To test a limit switch, you'll want to check continuity to determine what state the switch is in.


If there is a serviceman switch, turn it off. If not, shut the power off at the breaker panel.
Open the access panel and locate the high limit switch. It should be somewhere on the hot box.
Disconnect the wires from the switch (don't forget which goes where).
Using a multimeter, test for continuity between the leads on the switch.


There is an easier test to check limit switches, but it should only be performed by HVAC techs. So I will not discuss it here.

If the circuit is open, you'll want to replace the limit switch.

If the limit switch is adjustable, make sure it is set properly according to the manufactures specifications.

If the limit switch tests OK; or you've replaced it and still have the issue, you'll want to focus on the thermostat.

From your description, it sounds more like a problem with the thermostat.  Most safety mechanisms in furnaces do not let the furnace fire if there's a fault, so if you're getting flame I would not suspect a limit switch. I would suspect either the thermostat has gone bad, or it is miswired. If the fan is always on I would suspect either the fan switch has failed closed (is always in the ON position), or the fan wire was incorrectly bonded to the voltage feed line. 

Thermostats are basically switch boxes. They'll have one live wire [R] (usually 24v in the US, or line voltage in other parts of the world), one heat call wire [W], one blower fan call wire [G], and optionally a cool call [Y], and/or C wire. When the thermostat wants to warm up, it will connect the R to W and G which tells the furnace the thermostat is cold and needs heat. If it's hot it connects the R to Y and G, to tell the furnace it's hot and wants to be cooled down. If you have a FAN control and you turn it to ON, the thermostat connects R to G and the blower fan turns on.

The easiest way to rule out the thermostat, is to replace it.  Troubleshooting thermostats is a pain in the ass, so it's much easier to simply switch it out.

If you've changed the thermostat and you're still having the problem, call an HVAC tech.