Drama comes from making decisions with consequence. Hit points suck balls at making this happen.
If you ask different people what hit points mean, they say different things. What does losing HP even represent? When do you start bleeding? Does HP represent the quantity of blood in your bones or just a numerical measure of fighting spirit/stamina? How does healing magic work? Does it bolster your spirit or actually cause your physical wounds to knit shut? How much of HP is psychological and how much is physical? Does HP work differently for characters and monsters? When is something bloodied?
A large percentage of games us this mechanic, but a lot of them fail to clarify what they actually means. The abstract needs to exist for gameplay to happen, but it’s just oddly unsatisfying. The fact is, RPGs require something to show how not dead your character is. If characters died immediately when something hit them, that would suck as fights would be swingy, but detailed lists of bruises and injury charts would be complicated and slow the game down.
Playing D&D or its variants requires you to live with the weirdness of HP. It lives in every TSR, WoTC or 3rd Party book, and most OSR books keep it as well. It can’t be avoided if you don’t want to mod the shit out of everything you read.
All of the stuff above is irrelevant, really. You can handle it just by talking about how you deal or receive damage with different words and bits of imagery. The bad thing is that hit points suck the consequences out of RPGs.
Your HP regenerate when you sleep overnight. This means that even you are on deaths doorstep, you can fully recover from whatever maladies you have just by taking an 8 hour nap with a sleeping bag (and maybe rollmat) in the dank and dangerous gloom of a poorly lit cave with no fire and no food. This means that combat in the dungeon does nothing, as long as you can take a rest, and the monster doesn’t fully kill you. Combat is no longer dangerous. You’ll be fine if you avoid death (1) and can somehow sleep.
If you want keep HP, you need to find new ways to inject consequence into your game. You players will not properly respect danger if they can bounce off whatever ills you throw at them. Make them afraid!
Maybe you could add an injury table or an insanity table. My game, Best Left Buried, has both of these, and will be available soon. Make your player deal with the risk of permanent dismemberment or disability if they get crit or are made unconscious. If they face and eldritch horror and only just win, give them an irrational fear or compulsion, or start mucking around with the Trait/Ideal/Bond/Flaw on their character sheet.
The other way is to force them to deal with consequences that are material or emotional. Give them something or someone they care about, and risk it being taken away from them if they make a mistake. If you threaten their favourite NPC or settlement, they will take that seriously. If someone tries to take their magic item or gold, I bet you they will start to care.
The final way is my favourite. Make the rest do different things depending on how good a night sleep they got, looking at variables like light, food, comfort and interruptions. For example, few hours of sleep scratched together in a wet, damp cave without a fire, while monsters hunt for them, might grant no HP at all. Six hours of rest with a single watch shift in a relatively dry cave, along with some rations and a small fire, might restore a few hit dice worth of HP. If you manage to get a full night sleep in a tavern or a well guarded base, you get all your HP back. This adds a new element to your game, where establishing a well guarded camp with a good supply of food, light and warmth is vital when journeying into multi-rest dungeons.
If you give them risk that matters, then drama happens. Drama makes the players care more. Try it out in your normal game. It’ll work. Honest.
(1) At higher levels, death doesn’t really even slow you down either. This is another sapping of consequence away from the game, but is a fun end game reward and should PROBABLY be allowed.