For general maintainability, I recommend avoiding $word->ActiveDocument. If you'll only be working with the current document, then do my $doc = $word->ActiveDocument somewhere at the top of your code.

Going from this Stackoverflow answer, you could use $doc->Selection->GoTo as suggested in the Microsoft documentation:

for my $para (1..$doc->Paragraphs->Count) {
    $doc->Selection->GoTo( What => wdGoToParagraph, Which => wdGoToAbs
+olute, Count => $i );
    my $para = $doc->Selection->Paragraphs->[0];
    $doc->Selection->GoTo( What => wdGoToTable, Which => wdGoToNext );
    my $table = $doc->Selection->Tables->[0];

    ...
}
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This will still trip for paragraphs that have no table. Maybe in that case, it makes sense to step through all tables instead and then go backwards to find the corresponding paragraph or heading, depending on how your document is structured.

It turns out all the info I need is in tables. I can do something like:

 my $tables = $word->ActiveDocument->{Tables}; 
    for my $table (in $tables) {
 
        my $text = $table->ConvertToText(wdSeparateByTabs)->Text;
.....

                                  }
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The question is: instead of getting the content as a blob of text, can I get it per row and column?

Again, from the Microsoft documentation on the MS Word Table Object, look at the ->Cell method. With that, you can get the content of a table by row and column.