First, I should state we do NOT use CON modifier for bonus HP when you level. We play you add your highest ability modifier to your HP at 1st level only (we want a lower HP game). This is true for creatures as well as PCs.
What we are doing is folding levels of Exhaustion into CON after a lengthy discussion last night, and we'll try this out next time:
Exhaustion (replaced by CON loss)
Levels of exhaustion are replaced by reduction in CON. Each level of exhaustion is instead a reduction of CON by 1 point.
Since you regain 1 level of exhaustion per short rest, you regain 1 point of CON instead.
Unless you roll for ability scores, you won't have a CON less than 8 to begin the game. Since it isn't linked to HP via a per level bonus, this works for us. Also, I checked my creature database and only Gas Spores (that I could find) had a CON below 8.
This allows most creatures to take exhaustion before feeling the effects, which begin at CON 7. However, since CON is still a common save and used in other ways, you don't want a high CON to get low or you'll lose your modifier.
Critical Hits
You do not score a critical hit on a roll of 20. Instead, whenever a damage die rolls maximum, it explodes and is considered a critical. You roll another die and add it to the maximum roll, continuing until you do not roll maximum. This applies to weapon and spell damage.
Features which grant you additional damage dice can also explode on a maximum roll.
Dropping to 0 Hit Points
When your hit points are reduced to 0, any remaining damage is applied to your CON score. You are stable and make a DC 20 CON save to remain conscious. If you fail the check, you are "knocked out" for a number of rounds equal to the amount of CON reduction you have.
If the damage which reduced you to 0 hit points was critical, you are
not stable. You lose 1 point of CON at the end of each of your turns until you become stable. You also make a DC 20 CON save to remain conscious, but with disadvantage.
If you take damage while at 0 hit points, it is deducted from your CON score and if you are conscious you must make another check to remain conscious. If damage is critical, this save is also with disadvantage.
Stabilizing a Creature
A creature can self-stabilize by making a DC 20 CON save at the start of its turn.
You can also make a Wisdom (Medicine) check to stabilize a creature. The DC is equal to 10 + CON loss, and expends a use of a healing kit. If you do not have a healing kit, the check is made with disadvantage.
Regaining Consciousness
An unconscious creature will regain consciousness in 1d4 hours + 1 hour for each point of CON lost, and regain 1 hp when they become conscious. You can make a Wisdom (Medicine) check, DC 10 + CON loss, to revive an unconscious creature, by expending a use of a healing kit.
Magical Healing
Spells and magic which restore hit points do not help in restoring lost CON. Lesser and Greater Restoration spells will restore 1 point of CON for each spell level used when the spell is cast.
A potion of Vitality will restore 2d6 points of lost CON.
Natural healing
You regain 1 point of CON per long rest up until CON 10, and then 2 points per long rest when you are above CON 10. For example, if you have CON 14, but it is reduced to CON 8, you will require two long rests to get it to CON 10, and then two more long rests to restore it to CON 14.