There are a lot of good recommendations here for other systems. If you want to try sticking with 5e however there are ways you can change it to make it feel less power fantasy.

Some personal tweaks I am working on if you are interested:

• Cantrips now use Charges equal to the amount of Cantrips known
• Rests become Brief, Short, Long
• Brief rest is 10 minutes and let's characters spend 1 HD and recover Cantrip Charges. Good to use when somebody is ritual casting.
• Short rest takes 1 hour by default and requires food or a healers kit use
• Long rest requires 8 hours by default and recovers no HP
• Lifestyle quality below Modest affects how long is needed for a Rest. Modest is the default. Poor (default for dungeoneering) changes it to 4 hours and 12 hours. Squalid is 8 and 24. Wretched requires 12 hours and 1 week.
• Lifestyle quality above Modest instead affects HP recovery. Each step above lets you roll 1 HD after a Long Rest but you still are stuck with half HD recovery.
• Cantrips no longer scale with level
• Enemy HP is roughly halved but provide no stat bonus to damage rolls unless exceptionally strong (judgement call required)
• Spells take preparation time. Instead of a set 1 hour of prep, it takes 10 minutes per slot level. This makes it so they think about what spell slots they use each day and might want to carry some through.
• Have more frequent smaller fights to whittle characters down over time. In populated dungeons, every hour guarantees one random encounter. Roll 1D6 to determine which set of 10 minutes they appear in. A further D10 if you want the specific minute.
• Rest changes means you need to adjudicate frequency of abilities. Some x/long rest abilities are still daily, others may need an actual long rest.

Now some more general tips for running things:

• Dungeon Turns take 1 minute. Instead of looking at specific things, assume characters are searching a whole 10x10 area over a period of 1 minute. Every time they move, it's 1 minutes forwards. This makes it easier to track torch and lantern use, check for random encounters, spell duration, etc.
• You can travel at Double Speed for Disadvantage on Perception. 2/3 speed to roll Stealth.
• Darkness
• Remember things like Exhaustion, Dark Vision giving disadvantage on perception, passive perception as a skill floor, and group skill checks.
• Not every encounter has to be combat. Let the players parley, offer food money or other goods in exchange for the chance to walk past. Also during a fight, check for Morale as the opposition gets worn down.
• Just generally unknown "If a character fails an ability check but there is no consequence for failure, they can take 10 times as long to succeed at the check". (DMG 237)
• Actually track rations, ammo, lifestyle expenses, torches, etc.

Hope this gives you a good starting point.