Yes, the discomfort of seeing a doctor is better than losing your fingers.
I agree with Jemma, it sounds like an infection that needs to be seen by your Doctor, as you probably need antibiotics either by local application, or oral medication at this point.
I would advise you to see your doctor as you have an infection in the fingers and this needs to be treated by an antibiotic either by tablet or an antibiotic cream. Infections can spread, so it is important for you to see the doctor. He is used to seeing people with all sorts of problems, so don't worry about it.
You could try bathing the fingers in some warm water with salt in it or some Bicarbonate of Soda.
I would put in about a tablespoonful of salt to a medium sized bowl of water.
Don't pick the nails, keep of the plastic glue, nail varnish and false nails. Make sure you keep your hands clean. You could also apply an antiseptic cream like germolene or one that your pharmacist recommends.
Best wishes.