I'm sorry to hear that. The 5E rules are really working against you there. Do you have some sort of house rules, to combat the fact that it's impossible for anyone to survive losing more blood in a fight than they are capable of regenerating overnight?
Personally, I prefer players to really feel it when they get hurt. When your fighter gets stabbed, you should know that something terrible has happened, and the player should start panicking. That generally means that the PCs shouldn't be getting hit in every combat, but that hits which do land should deal at least a third of their total HP, which will not recover overnight. (Again, though, the 5E rules are really working against me on that one. The default rules really favor using lots of minions to bloodlessly pummel the PCs into exhaustion over the course of a long day.)
I am still experimenting to find an ideal mechanic. But one thing is certain, that hitting zero hit points hurts. Those are the lethal wounds, and if survived, leave a scar of some kind.
Where zero hit points is a threat to ‘life and limb’, it may result in physical damage rather than death, such as the impairment or even loss of a limb − at least a fatigue-like condition.
I also want to separate Constitution bonus hit points ( = meat points) into a separate pool from all other hit points ( = nonphysical skill/energy/luck points). Then the Constitution hit points take longer to heal. Here the system works better 4e style, where the Constitution score is a one-time sizable bonus without later additions while leveling.
I am still planning future experiments. I want a simple-as-possible elegant mechanic that hits all the right possibilities that feel realistic.
That said. Normal, nonlethal damage (before reaching zero hit points) feels more realistic when it is mainly superficial damage that heals easily, because it comes with no mechanical conditions or consequences. It is only the zero hit points that I feel the players should fear.
Heh. It just occurred to me. Maybe bandaging a wound to heal hit points, should instead be temporary hit points that exist only as long as the bandage stays on. If these temporary hit points get damaged, it means the bandages are ruined and the wounds need to be re-treated. The characters wear the bandages for as long as it takes to heal naturally, while real hit points replace the temporary hit points. The need to remember that the characters are still wearing bandages (and balms and continually cleaning wounds) helps the players feel that characters did get injured.
Anyway, like I said, I am still experimenting to get an injury system that works well for me.