Or could I put the module's code in the same directory as my Perl code?

Yes, if the module is in the same directory, you can just do do this, presuming the module is called Whatever.pm:

use Whatever;
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Notice that is not it's normal canonical name (eg, Special::Whatever). You should test this way to be sure it will work with any particular module, of course.

If you put it in some other directory you can do this:

BEGIN { push @INC,"/some/directory"; }
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And use "use" the same way.