woods - gramatically singular or plural

I believe the answer to your question should be "The woods near the town contain many secrets."

Cambridge Dictionary calls "woods" a "plural noun", with the example "Shaded from the sun, the woods were cool and quiet."

wiktionary.org is more cautious: "usually in the plural, sometimes singular" and "Woods more often takes a plural verb (determiner, etc, as in these woods are) than a singular verb (as in this woods is)."

There is an interesting and apropos comment on english.stackexchange.com:

It gets trickier when there's an adjective. Although I can find both of these constructions in Google books, I wouldn't say a small woods (because it's plural), or some small woods (because mass nouns don't work that way; you can't say some small rice either). I'd say a small stretch of woods or a large expanse of woods. – Peter Shor