Based on what you've provided, you can simply put your old and new URLs into variables and do the substitution the same way as you did with s/ABC/CDE/g. Here it is on the commandline:

ken@ganymede: ~/tmp
$ perl -Mstrict -Mwarnings -E '
> my $line = q{blah blah https://abc.net/cm/one blah blah};
> my $old_url = q{https://abc.net/cm/one};
> my $new_url = q{https://abcde.net/dm/two};
> $line =~ s/$old_url/$new_url/g;
> say $line;
> '
blah blah https://abcde.net/dm/two blah blah
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