If you have even a slight fever and feel ill, based on the info that it goes down your ring finger into your pinky I think you need to see a doctor. This could now be cellulitis. If by going down you mean it's red and itchy or tingly or painful and perhaps a little warm.
This is not a horror story. It's what happened to me and I think it's important for you to consider this possiblity and if any of my description fits for you to call an advice nurse.
I say this because I had a very small sore on my toe. Then I thought I had the flu and then my inner thigh was sensitive to touch. It didn't exactly hurt but it wasn't comfortable and it felt a little warm. I called the advice nurse who said I should immediately come in to Urgent Care. Yep, it was cellulitis that had gone from my toe to my thigh. Before this was over I had several large blisters on my toe. I was on antibiotics for a couple of weeks. I was not hospitalized because the redness and sensitivity had not gone into my groin. If I had not gotten treatment before it went into my groin I would have been in serious trouble.
You may just have a yeast infection. Yeast is a fungus, I think. In which case you need to use an anti-fungal medication, such as what you can buy OTC for athletes foot.
I also suggest that you leave your finger without a bandage for several hours/day at some time so that the wound can dry up. Constant moisture prevents wounds from healing. It needs to ooze. Ooze is what helps to get the bacteria/yeast out of the body.
Have you tried using rubber gloves? Even tho it's bandaged it does get wet while you're doing housework.