Hi, Yes, you have the answer to you're problem. Now, I don't no if you have just a high limit, or a combination Limit control? I will go with the combination fan control as when they go bad, they stick in the closed position and they have normally open contacts. When the heat exchanger starts to heat on a call for heat, there is a dial on the fan control with temp. settings on it , and it will start to rotate as the heat gets higher. When it reaches its set fan on temp, it with start the fan.We no the fan works so the blower is good, so it shuts the burners down on you're high limit temp. which is factory set to about 250*F. When you turn The t-stat from auto to on in the fan side, that solves the problem temp, only. You will need to R & R the Fan control. If you need some direction on doing this please let me know and if it is a Honeywell auto/on type that is rectangular in looks, and if you remove the cover you will see the temp. dial with arrows on it. Thank you.
Best of Luck, and Sincerely,
Shastalaker7