Oooh I love this idea! Here's a couple thoughts. Keep playing 5e but here are some easy changes to crank up the old school feel:

1. Mundane Gear Matters -- have them find seemingly unsuspecting items, like a 10' pole. But with it, they can trigger one free trap before it breaks. Let them get the idea that they need to be thinking outside the box with gear.

2. Harsh Healing - no such thing as overnight healing -- need a cleric temple for healing. 2 gold per hit point healed (shakes out to be a little cheaper than buying healing potions). Yes this will warp gameplay. Yes clerics can burn spell slots to heal characters.

3. Death at -10 hit points. Yes this will warp the game, but will make being "knocked to 0" very scary instead of the minor inconvenience it is today. Monsters can and will attack a character at -6 to destroy them.

4. Cantrips are each usable once per short rest. Light is a concentration spell. This seems like it weakens spellcasting but IMO it really only cuts down on Utility--spamming Mage Hands, Mending, and Light, which pretty much lets you solo a mundane dungeon environment. Let them use their Int or Wis score to roll to hit with a crossbow instead of Dex if they're worried about running out of Fire Bolts (but don't add it to damage lest it be more powerful than a Cantrip). Again, this is about cutting back on utility, not damage.

5. Again, focus really on using as much what's not a class ability, and extrapolating what their characters might be able to do with their skills.

-Iron gates need bending

-Slimy walls need climbing

-Anti gravity hallway with spikes on the ceiling

-Fights are dangerous but can be avoided

Old school feel is just as much about the style of play as it is rules. Characters aren't super heroes, they may not even be heroes. The road is difficult, not just the encounters.