Hello, "I'll give my life for you" is rather a literal translation of the Romanian text. The meaning is more about needing someone's presence, the person would die without the lover's presence, and so on. A suggestion would be: "I so long for you", "I so want your presence". It is a Romanian expression that does not have an ecquivalent in English really, but it sure isn't the meaning it has been given by the translator here.
Now I saw the discussion. If the English translation was done according to the Romanian text, than "I'm dying to have you with me" is not correct. It should be "I'm madly in love with you!"
But the source language is not Romanian, so the translation should be done according to that.