1) I poured too much engine oil into the engine and caused my car to give off/emit a lot of white smoke.
2) My car give a final burn off/out of smokes when I was driving home. My car is back to normal now. (burn off/out= I wanted to say the final emission of a lot of smokes).
Are the above sentences correct? Please advice.
In 1), you don't need to use "engine" twice. It's understood that you put engine oil into the engine. Both "give off" and "emit" would be correct.
In 2), you must use the past tense ["gave"] and the singular of smoke. Even if it did it more than once, then it would be "bursts of smoke"
I'm not a teacher of English, but I have spoken it for (almost) all of my life....