Before running the query you could set the case_sensitive_like pragma to ON.
PRAGMA case_sensitive_like = boolean;
The default behavior of the LIKE operator is to ignore case for ASCII
characters. Hence, by default 'a' LIKE 'A' is true. The
case_sensitive_like pragma installs a new application-defined LIKE
function that is either case sensitive or insensitive depending on the
value of the case_sensitive_like pragma. When case_sensitive_like is
disabled, the default LIKE behavior is expressed. When
case_sensitive_like is enabled, case becomes significant. So, for
example, 'a' LIKE 'A' is false but 'a' LIKE 'a' is still true.
This pragma uses sqlite3_create_function() to overload the LIKE and
GLOB functions, which may override previous implementations of LIKE
and GLOB registered by the application. This pragma only changes the
behavior of the SQL LIKE operator. It does not change the behavior of
the sqlite3_strlike() C-language interface, which is always case
insensitive.
Then just run the query as it is.
PRAGMA case_sensitive_like=ON;
select username from users where username like 'aDam%';
Test it here on Sql Fiddle.