How to use weights from our data set properly? Should we use them at all?
Firstly, if a variable is imbalanced (Race in your case), you shouldn't worry about. As its a categorical variable, it will be converted to two or more dummy variables.

Secondly, if the weight represents the number of people for each entry, as the following example, you should include the weights. 

hypothetical data

Race       Education    Marital    weight
------------------------------------------
White       tertiary    Married      10   
White       tertiary    Single        6
Black       tertiary    Married       8
White       secondary   Single        4
...
...


For this example, I would oversample each entry by its weight. Because your real database is:

          X               y
---------------------  --------
Race        Education   Marital
-----------------------------------
White       tertiary    Married   
White       tertiary    Married
White       tertiary    Married   10 records
...
White       tertiary    Married

White       tertiary    Single 
White       tertiary    Single     6 records
...
White       tertiary    Single
Black       tertiary    Married
Black       tertiary    Married    8 records
...
Black       tertiary    Married
White       secondary   Single
White       secondary   Single
White       secondary   Single
White       secondary   Single


However, I should see few records of your dataset to help you better.