Something very similar happened to a friend of mine, sick for 3 days but it did go away.
Hydrated, heathy, energized, took pills

After a day of not drinking or eating, my sickness is now gone. Odd that medicine didn't cure it but just had to wait it out.

So I've been through taht circle now for three times:
Got infected through stiching myself up.
All the messages through "itchy wounds" and so on. After some time getting hot, finally wounds look cleaner and going back to healing, and all green statuses. healing never leads to healthy!
As soon as I dring from my canteen (containing purified water) after the sick status is gone and the healing status is there i will get the sick status again and go through the whole circle.

Cleaning the wounds didn't help. I used two bottles of alcohol up to now.

Got sick from eating rice, caused infected wounds. Buddy didn't believe me, so he ate the rice too, he also got "Sick" status and "Your wounds are itchy". Powering through all 4 stages until your wounds feel clean does not make the "sick" go away... and sooner or later, drinking/eating anything will cause infected wounds all over again.

Update: I packed all my gear into a Barrel, and suicide to clear the sick. Played several hours just fine, sticking to newly found food and water. Worked my way back to barrel. Took PET bottle to pump, emptied bottle, refilled at pump, and used water purification tab. Drinking from PET bottle caused instant SICK and "Your wounds are itchy".

Update 2: Food/water items are getting tainted with some mystery disease that's causes "infected wounds". Forcing vomiting cures the "Sick" status and alcohol tincture fixes the infected wound. Ditching all water containers, Rice, cereal, and fruit prevented re-infection. Infected items looted off my corpse by other players made them sick too. Tried using disinfectant spray on known culprit PET Bottle, caused visual bugs replacing my "Energized" and "Hydrated" with "sick", forcing an HUD update by overeating until "stuffed" cleared the display bug... no text messages indicated itchy wounds or other ailments. Not sure what this disease is, I dumped out all water, and refilled from well, and used purification tabs... but PET bottle was still infectious.

Additional Details: Water containers that have become infected or corrupted (not sure what else to call it) will re-infect the player despite empty/refill cycles and water purification tabs. Additionally, just emptying a corrupted container on the ground will trigger a UI bug that shows your status as "Sick" and removes hydrated and energized status but does not appear to actually infect the player. Logging out and in fixes this UI bug.