It certainly sounds like a problem in which unicode is involved... but it might be better to call it a "mismatched encodings" problem. Whatever page is being pulled in by LWP, it is apparently using some sort of "smart" or "wide-character" variants for the quotes and dashes, and in order to do that, the page should be labeled as to the particular (non-ASCII) character encoding that it is using in order to represent these special characters.

Meanwhile, your own "current" web page is probably specifying a different character encoding, and/or you are viewing the page with a browser that is forcing its display to use some particular encoding, and the result is a conflict (a mismatch) with the original data received via LWP, so you are seeing what happens when the characters are misinterpreted.

It's also possible that your script may be doing certain "standard" operations on the data, via CPAN modules or your own code, and in the process, perl is doing some sort of "default, assumed-to-be-reasonable" conversion of the character encoding, again with the result that the special characters are being misinterpreted as something that they were not meant to be.

If you can show the original url, or some of the relevant unmodified strings from that page, and/or some minimal snippet of your own code that produces this behavior, it would be more likely that we could pinpoint the issue(s) for you.