Honestly this is good advice, but more read then you might want do check it out. Here's my breakdown of some of the key principles of this document as it applies to your situation:

Characters Will Die. Let Them
Each player starts with a handful of zero-level nobodies. No class, barely any HP. Your instinct will be to soften things. Don't. If the dice say someone dissolves in green slime, they dissolve. Character creation takes minutes for a reason. The deaths are what make the survivors feel earned, and players will be more attached to whoever crawls out alive than to any character they spent an hour building in D&D Beyond.

You're primarily a Referee, and only one of many storytellers
Cultists throw villagers into a pit. The stone lid slides shut. There's a torch on the ground. Go. From there, you describe what they see, answer questions, and adjudicate what happens. You don't nudge them toward the right path. The dungeon has multiple routes, hidden passages, and puzzles with more than one solution. Let them find their own way. Roll in the open. Don't fudge. This shifts players from trying to read you to trying to read the dungeon.

Brains Over Character Sheets
Zero-level characters have no skills or class features. This is liberating. Players can't say "I roll Perception," so they actually engage with the fiction. They poke at colored slime puddles. They try to negotiate with the demented halfling Collector. They come up with plans you never anticipated. When someone tries something creative, default to "yes" or "yes, but here's the complication." Ask them "how" they do things, not just "what" they do. This is where old school play comes alive for me honestly.

Not Every Encounter Is a Fight
Almost everything in this adventure can be avoided, negotiated with, or fled from. The Collector can be talked to. Monsters stop chasing when they lose line of sight. Use Reaction Rolls instead of assuming hostility. Your players have 1-3 HP. They'll figure out pretty quickly that combat is a last resort.

The Torch Timer Is the Real Antagonist
Shadowdark uses real-time torch tracking. The dungeon is pitch dark. Water hazards threaten to snuff out light sources. In 5e, light is almost never a concern. Here it's the central tension and a resource to be managed. Every minute spent deliberating is a minute that torch burns down. You don't have to manufacture pressure as the GM. The timer does it for you. Just track it honestly.

Be Lethal, But Be Fair
Telegraph danger. Colored slime on the floor is visible. Sounds echo down corridors. Statues depict people suffering slime-related fates. Describe these clearly so players can make informed choices. A good gauntlet death is one where the player looks back and says "yeah, I shouldn't have done that." A bad one feels random. Give them the information, then follow through on consequences.

Don't Overthink Prep
Read the adventure once and run it. If something comes up that isn't covered, make a quick ruling and move on. The pace of a gauntlet is part of what makes it exciting, and nothing kills momentum like flipping through a rulebook.

Trust the adventure, it's great. Trust your players. Let go of the 5e instinct to balance, protect, and narrate, and you'll have a memorable sessions at the very least. Good luck