I want the whole rows which satisfy that unique condition with all columns. I want to iterate in the next step.
So you don't want to group by that field and return one of the multiple rows. You want only rows which are unique.
One way is using Enumerable.GroupBy
and count the rows in each group:
var uniqueRows = dt.AsEnumerable()
.GroupBy(r => r.Field<int>("OrganizationID_int"))
.Where(g => g.Count() == 1)
.Select(g => g.First());