I have been toying with an idea (I hate whack a mole too):
You drop to 0hp you don't fall unconscious! You are still up! Huzzah! But...
1) Take immediate point of Exhaustion
2) Still make Death Saves: Fail = Add point of Exhaustion, 3 Fails = pass out (unconscious), Success = nothing, 3 successes = Stop making saves, lose 1 point exhaustion. Crit success = lose 1 point exhaustion and have 1hp. You keep making death saves until you get three successes (or a crit) or you die from exhaustion (6 exhaustion!)
3) Healing when at 0hp = each dice used (or each 5hp for effects like Lay on Hands) removes 1 point of exhaustion. Once all exhaustion is gone, then healing HP can begin (so 3 exhaustion is healed by a 4d8 cure wounds with only 1d8 delivering HP)
4) Any damage from a single source = gain exhaustion and lose 1 hd (or another exhaustion if no HD)
Probably need to call it "Wounds" not exhaustion so it doesn't heal with Barbarian Beserker rage-induced exhaustion (although they should stack for penalty)
something like that. first time I put on "paper" so its very rough mind download. You could make success death saves lose exhaustion as you swim in and out of effectiveness. Actually, if you are up and fighting, you are a legitimate target, so you could easily reach 6 exhaustion very quickly, maybe success = lose 1 point exhaustion is a better option.
It needs playtesting, obviously.