As per the comments, I'm going to assume you're getting mixed up between update and insert.

 USE [DB_A] ;
 UPDATE dbo.address
 SET City = 'test',
     State = 'ing'
 WHERE City IS NULL ;

And

 USE [DB_A] ;
 UPDATE dbo.address
 SET City = NULL,
     State = NULL
 WHERE City = 'test' ;

I think you're there otherwise :-)

INSERT INTO atable (x,y,z) VALUES ( NULL,NULL,NULL)

They are different. When you enclose something in single quotes, it becomes a string -- this is true of all keywords in SQL, including NULL.

When you use just NULL, it is the SQL keyword, and it is compatible with all types.

So, these are quite different:

insert into t (col1, col2)
    values ('2021-05-25 11:58:41.000', NULL);

and:

insert into t (col1, col2)
    values ('2021-05-25 11:58:41.000', 'NULL');

'NULL' is a string and only compatible with a string type. You should get a type-conversion error if the column is not a string.

The one caveat is if you are storing the values in a text file for loading into the database. In such a file, strings may or may not be delimited with single quotes, depending on the structure of the file.

When you use a column with NOT NULL then it's not possible to add NULL to the column.

I guess you have other columns as well, like:

Create TABLE Products
(
[ProdID] [int] IDENTITY(1,1) NOT NULL,
[Col2] int NULL,
[Col3] int NOT NULL,
[Col4] int NULL
)

When you want to insert a record to this table dont mention the [ProdID] column, it will be filled automaticly because its a identity.
Example:

  INSERT INTO Products ([Col2],[Col3],[Col4]) VALUES (1,2,NULL)