I would like the output to be as follows: What am I doing wrong?

Why would you like the output to be as follows? What are you actually trying to accomplish?

Is this a test/quiz question?

Or

What book are you learning from?

Have you read require?

This is how you should write that program

my $database = LoadDatabase('a.pl');
print "$database\n";
$database = LoadDatabase('b.pl');
print "$database\n";
$database = LoadDatabase('a.pl');
print "$database\n";
exit( 0 );
sub LoadDatabase {
    ....
}
[download]

Devel::Trace will trace your program exactly

>> c.pl:1: require "a.pl";
>> a.pl:1: use strict;
...
>> a.pl:2: use warnings;
...
>> c.pl:2: print "$database\n";
>> c.pl:3: require "b.pl";
>> b.pl:1: use strict;
...
>> b.pl:2: use warnings;
...
>> c.pl:4: print "$database\n";
>> c.pl:5: require "a.pl";
>> c.pl:6: print "$database\n";
[download]

As you can see the last require does't execute a.pl because its already loaded