"Cosset" is a word for a pet lamb.
Lambie is a sort of made-up word like lambkin. You might use it for example, when calling to a lamb to come to you. Or as a term of endearment to a person, like saying "honeybun" or "sweetheart". It's not what I would call a proper dictionary word.
I wasn't aware that "lame" meant an old-fashioned person.