No limp mode, It migh be the temp sensor is bad. You can check the temp sensor by removing it and using a multi meter to see if you have continuity from the connector plug on top to the bottom. If you have continuity when it is cold, the sensor is bad. If you have no continuity you can test it by putting it in a pan of water and bring the temp up to about 205*f. Carefully, wearing gloves or a pot holder check to see if you have continuity from top to bottom. The sensor is just a heat sensitive switch that makes contact as the switch expands to complete a circuit in the sensor, causing the switch to allow ground to the cylinder head.
If you touch or ground the brown wire to the cylinder head it will cause the beeper to sound too, kinda like the temp sensor does when it gets hot from being over heated as if you had a cooling problem. It sounds like it is the sensor that is bad. You don't want to run it, and chance a over heat condition. I hope this explains the heat sensor and it solves your problem.
Karl
