You have retained a lawyer who is presumably competent so you should first of all ask him. There may be elements in your case that I am unaware of that would affect an answer. In general a single DUI is not a crime of moral turpitude although a series of DUIs may give rise to an inference that you are an alcoholic which would have immigration consequences. Driving without a license is not a crime of moral turpitude either. You should consult further with your retained attorney regarding these issues. If s/he doesn't know the answer you should replace her/him.
Your DUI attorney for court SHOULD be keeping you informed not only of immigration issues ( which based on what you have said should not be an issue) and how to address the license issue, if they know what they are doing.
First let me say that I agree with the previous answers by Mr. Kaman and Mr. Geller.
Given what appears to be an unfortunate lack of communication between you and the lawyer you hired, it is understandable that you are concerned about what will happen at your court hearing (presumably your arraignment, i.e., your first appearance). You should know however that substantive decisions impacting the merits of a charge or the consequences of potential convictions are rarely at issue at the first court appearance.
Daniel A. Flores