You might try testing the resistor on the led to get the actual value and change it out for a resistor with a slightly higher or lower value, maybe 20-50ohms in both directions to see which helps the most.
You may have to give up a little overall brightness to be able to get the led to turn off completely.
Although ohms law dictates how they should work, sometimes it's best to try stuff to see what works best. About the worst that can happen is you burn out a handful of led's.
I never trust any pre-wired led myself, I take a sample from each batch of led's I get & test the actual voltage unless I have a detailed spec sheet on them... and even then I test them. That way I know the characteristics of each led & how it should react.
One thing to note is that although vehicles are considered 12v, when running the alternator can produce 14.5v or more which can totally mess with your calculations.
That's why some of my single led mods may be a little less bright when the truck is off (only getting 12v+/- battery voltage), but when it's running (full alternator voltage of 13.5v-14.5v), they get full voltage & really come to life.
The warning beeps and/or dash indicators should indicate that the bulb is burned out, or that the led's resistors aren't correct. If you run a resistor across the leads, it will basically see it as a short or open circuit depending on the value of the resistors & thus give the warnings. I think once you figure out the resistor value, you should be good to go.
It's been a while since I started doing that mod... but my info probably wouldn't help as I'm sure the led's I use have different specs than the ones you're using.