This is what the law says:
Exclusion for Physical or Mental Disorders
Under INA §212(a)(1)(A)(ii), aliens are excludable if they have been determined in accordance with the regulations of the Secretary of Health and Human Services in consultation with the Attorney General:
to have a physical or mental disorder and a history of behavior associated with the disorder that may pose or has posed a threat to the property, safety or welfare of the alien or others; or
to have had a physical or mental disorder and a history or behavior associated with the disorder that may pose or has posed a threat to the property, safety or welfare of the alien or others and which behavior is likely to recur or lead to other harmful behavior.
This ground does not specifically refer to alcoholics or alcoholism (it did prior to 1990). Nevertheless, according to Note 6.3 to §40.11 of the FAM, alcoholism constitutes a medical condition. Evaluation for alcohol abuse or dependence is included in the evaluation for mental and physical disorders with associated harmful behavior.
An alcoholic is not ineligible to receive a visa unless there is current, or a history of, harmful behavior associated with the disorder which has posed or is likely to pose a threat to the property, safety, or welfare of the alien or others. A conviction for driving while under the influence of alcohol has been determined by CDC to constitute evidence of medical disorder with associated harmful behavior.
In light of the above, it is possible that your two convictions could render you excludable under this ground. Of course, the USCIS would have to believe that your behaviour was likely to recur or lead to other harmful behavior before you would be found excludable.
If there is an alcoholism problem you should discuss this with an immigration attorney. There is a waiver you might be able to get, for this you probably would have to show some kind of successful treatment.