Then I strongly recommend not house-ruling things where you don't need to. Just run it as-is, and rule where gaps come up. With your first experience DMing, I do not recommend toying with game (re)design.
I think you'll find that this happens as people get low in hp anyway. They know how much damage they're taking per hit, and how many hp they have left, and therefore how many more hits they can take.
Oh, and as a recommendation for RP-encouraging mechanics, take a look at the rules for Inspiration. Players respond a lot better to rules that reward them rather than rules which threaten/punish them.