$sth = $dbh->prepare("SELECT username, password FROM users WHERE username =$username and password=$password");
my $sqlStatement = "SELECT username, password FROM users WHERE username =$username and password=$password";
print "Submitting: $sqlStatement\n";
$sth = $dbh->prepare($sqlStatement);
See what the full SQL statement looks like; you are likely to find an error there, since that's what your error message says.
Most SQL requires strings to be encapsulated in single-quotes: ... and password = 'newpassword';
Dunno if that's your problem, but the above changes should help shed light on what you are getting wrong in the SQL statement.
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