Do I need medical attention for my self harm wounds?

I cut with a clean blade 6 times horizontally on my arm when I was drunk last night and although most of them now just look like red scratches, one is gaping about 2mm, maybe 3mm and inside looks greyish (there's no puss or oozing so maybe it's just heeling?). I cleaned them and then dressed them. The cuts are each between 2 and 3 centimetres long. I had bad diarrhoea after eating and feel a bit lethargic and queazy but have no temperature or raised heart rate or anything, so is it safe to say I just have symptoms of a hangover and I do not have an infection from the cuts? I feel quite scared about it. I also feel really guilty.

Just to add - I would like advice on the physical side of this only... I already have mental health support in place and people are aware of my depression. I would just like to know whether what I've described is physically dangerous i.e. Infection?