RPing your character's health can be challenging. Especially in circumstances where your health is low but your fatigue is high. With this in mind I thought it would be a good idea to consider some RP aids.
My first thought is to have your maximum possible fatigue capped based on your current health. By fatigue I am referring to the part that goes like "physically rested" as I know that health already impacts recovery. For example, if you are "barely clinging to life" then your maximum fatigue level might be "physically exhausted". You can simply never recover beyond this until your health level increases.
From what I have seen, fatigue level impacts what my character can do more than anything else. When I am extremely low on fatigue my actions are limited and I think that it is reasonable that someone in 'critical condition' be severely limited. I might suggest a delay between reaching a health level and applying the fatigue cap though to represent the effects of adrenaline though. This would help prevent an increased death spiral during combat as the cap would likely hit after the fight is wrapped up.
For a less invasive option, maybe have regular notifications pop up like they do for diseases. Something to remind you that you shouldn't be touring the town while in "critical condition".
Of course, I don't claim to really know the health/fatigue system well enough to make a really good suggestion. I just wanted to throw out the idea is all.