So as stated by others your resistors wattage rating is likely too low for the wattage of the LED lamps, even if the LED lamp is 5 watts and you use a 5 watt resistor it may create a lot of heat. If you oversize the resistor to say a 10 watt it'll do the same thing with less heat.
Think of electrical stuff the same as you would a town waters supply. The wires are the pipes, the voltage is the PSI in the system, and your in line resistor is functioning as a valve. Using resistors in line to reduce voltage makes them act like a partially closed valve in a water pipe, the higher the Ω value of the resistor = the more closed the valve is on the water pipe. Now the voltage is pushing electrons through the resistor to the device, and the higher the flow through the resistor the hoter it gets (this can happen with under sized valves in high pressure water systems too).
Now with water you can just let it flow out of the valve and be done, but in electrical circuits you have to size the resistors capacity to handle flow (wattage) to the demand of the device which you are powering. So it sounds like your resistor is not rated to handle the flow it's experianing, with a multi-meter set to DC amps you can figure out the draw of the lights then back into what wattage resistor you need, I'd suggest sizing it at a factor of 2 (soI'd choose a 10watt resistor for a 5watt LED bulb).
WITH ALL THAT SAID, you're not really handling the issue correctly. You are trying to filter out a small spike in voltage that happens over the course of a couple of milliseconds that is coming from your turn signals LED bulbs when you cancel them. In electrical circuits, we call this noise, and its common for that noise to bounce through circuits with shared grounds (brake lights) and cancel the cruise control.
You really should be adding a simple Passive Low Pass Filter, which is what they use in stereo installs all the time to properly tune tweeters mainly. With an inline resistor and a capacitor bridging from between the LED bulb and resistor positive wire to ground. The resistor is still functioning as a valve, but the capacitor is what provides the absorption of the spike.
Let me explain using the towns water supply as an analog again. The water is supplied from the water tower and runs through a valve, then after that, it halfway fills a small tank before going to the end user. Well if there is suddenly a short yet high-pressure reversal of flow from the end user, the tank will act as a buffer and that pulse flow won't shock the whole system by trying to return to the water tower. How does this relate to electronics? let's recap; the water tower = the battery, the valve = the resistor, the tank between the valve & end user = the capacitor.
I put this type of filter on my e30 when I did my "one touch 3 blink mod". The circuit looks like this (keep in mind you have to size the resistor & cap correctly but its not that hard):