You’re somewhat confused, which is understandable since intellectual property rights and D&D is a confusing issue.
The OGL is a specific license with specific terms. D&D 3e/3.5e was made open for others’ use under the OGL and the open portion was published as a SRD, or System Reference Document. Other games derived from the d20 SRD (like Pathfinder and Mutants and Masterminds) and totally unrelated games, like FATE, use the OGL. 4e used a non-open license called the GSL, and previous versions had no available user license except under contract with TSR. You can read the blog post Open Gaming for Dummies to understand these very specific legal terms.
There are some “retroclones” that use the OGL and actually pull the OGL 3.5e information from the d20 SRD into something that looks more like an older version of D&D. See also What content can I reproduce from Pathfinder? that has a similar discussion specific to Pathfinder. Some people use the OGL to put out modules for various other non-OGL D&D versions, but that’s actually somewhat complex and you need to understand the rest of what’s going on in IP land to know how and why.
Wizards has also released 5e – or at least some of it – under the OGL as well. You can download it and the SRD on their site. Be careful, the SRD only has a subset of the game’s content, and it’s all that can be used under the OGL.
Let’s talk about using game IP in other games outside the scope of the OGL. The more general discussion in Can I use existing game mechanics in my own designs? has a lot of relevant details for you here, as well as Is it legal to “use” (reference) copyrighted material in a way that requires ownership of its original publication? You have three main areas of IP concern to contend with – copyright, trademark (including trade dress) and patents.
First of all, “fair use” does not apply in this case. At all. It’s one of those phrases people like to use on the Internet but don’t understand. If you up and use text from another work in your work, and it’s not part of the very limited scope of fair use (educational, review, journalistic, etc.) it’s illegal.
Secondly, game mechanics cannot be copyrighted. “Copyright does not protect the idea for a game, its name or title, or the method or methods for playing it. Nor does copyright protect any idea, system, method, device, or trademark material involved in developing, merchandising, or playing a game.” (United States Copyright Office, “Copyright Registration of Games”).
However, “the text matter describing the rules of the game” may be protected by copyright “if it contains a sufficient amount of literary or pictorial expression”. For example, a passage describing character creation for the Clans of the Sun and Moon, explaining their society and why they tend to have the skills they do, would probably be protected by copyright. A section merely describing the steps involved in rolling a number of dice or expending a number of points on X attributes would probably not be. (rpglibrary.org) Note that this means even charts and tables can be copyrighted. Everything’s copyrighted, there’s not a registry or list or anything (well there is, but stuff not on it is still copyrighted).
In general, this is the escape clause some people use to put out supplements/modules that simply have “goblins” and “skeletons” in them which may have “HD: 2+2” or “a DC 12 Perception check” outside the scope of the OGL. If it’s “mechanics” and not “expressive” that’s legal. But head onward and read about trademark…
However, various aspects of the game rules may also be trademarked. Several D&D monsters are, and in general things that would usually be Product Identity in an OGL game can be. Characters, game worlds, etc. Without a more specific license allowing it, you absolutely cannot set your module in the Forgotten Realms and have Elminster fighting a beholder in it. The reason many of the retroclones do use the OGL is so they don’t have to fret over whether somewhat unique terms like “Hit Dice” end up getting trademarked or not, since as a term you’re explicitly allowed to use it if you are using the OGL.
You will note that many products/publishers, out of fear of trademark won’t say they are “compatible with Dungeons & Dragons Fifth Edition” but instead say circuitous things like “for 5e!” Normally, expressing compatibility with something is an OK use of trademark, Hasbro themselves lost a lawsuit on that v. RADGames about a company claiming Monopoly compatibility for an add-on pack. But the OGL giveth and the OGL taketh away, the OGL says you can’t claim compatibility in its terms so those using it can’t legally say “D&D”.
Note that trade dress also means you can’t use graphics/art/logos/etc that make your product look “too much like” products from an existing publisher.
You can do trademark searches to look for trademarks, see the discussion in Are the names of the more generic planes copyrighted under the OGL as Product Identity?
Game mechanics can be patented, since processes can be patented. I do not believe there is a patent on file for RPGs or D&D but I’m not an IP lawyer. You can do patent searches as well. At least one RPG publisher applied for a RPG patent, check it out here (article on it). Just wanted to mention this for completeness.
Dungeon Masters Guild
Speaking of more specific licenses, Wizards has come out with one! Their “Dungeon Masters Guild” program is designed to specifically allow publishing of 5e material with a number of restrictions, primarily that such content may only be distributed through their online storefront and they get a 50% cut of the sales (though you can make it free – although you can’t distribute it other than through their site). The upsides are that (a) you can use the Forgotten Realms as a setting and (b) you can use all of5e rules/content, not just the subset provided in the SRD.
Bottom Line for Modules
“I am not an IP lawyer and this is not legal advice blah blah” but in general you have three paths open to you. All require a nontrivial amount of legal understanding, so taking a RPG.SE answer and going forth to publish would be the height of folly.
But the first is to use the OGL, limit yourself only to things found in the SRD, and don’t claim D&D compatibility. Most folks do this, for example Frog God as you note. For an adventure, with a little oversight, you are probably fine with just writing general scenario content with pure rules references.
Or, you can not use the OGL, navigate the copyright/trademark waters yourself (and/or hope Hasbro doesn’t care), and even claim compatibility. KenzerCo did this with some products; their owner, David Kenzer, is a lawyer so he knew what he was doing and figured he could hold his own. Since modules usually don’t contain large amounts of rules content with the exception of stat blocks, the problem space is pretty narrow.
Now, profit vs it’s a freebie on the Web doesn’t change most of the legal factors, but it does change immensely the likelihood you’ll get in trouble over it, so if you’re not really looking to publish for sale, go ahead. If you’re starting out and looking to publish for sale, it might be a good idea to do that through an established third party company (like FGG) that can help you navigate these waters.
Or, you can use the Dungeon Masters Guild program, if it suits your needs.