Just in the context of having answered Anonymous Monk above and reformulated my question, I found the origin of the problem.

While trying to solve my first problem (the path to the module not being recognized), I made the silly mistake of changing:

use VMS_utils;
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to

use VMS_UTILS;
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thinking that I would thus give Perl a better chance to find the module (VMS file names are all uppercase). But that was a mistake, because in my module I have:

package VMS_utils;
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So what happens with this, is that, indeed, Perl is able to load the module successfully, but it no longer recognizes the package, because it is declared with some lower case letters within the module and I invoked it with all upper case letters in the program.

Sorry to have bothered you with my silly mistakes, dear fellow monks, I should have found by myself, but I guess my mind was just too confused yesterday evening after having tried at least three dozen alternative syntaxes.