I used to cut when I was younger, I managed to stop myself from cutting but it led to another form of self-injury (I became bulimic) which I managed to overcome aswell. Recently I cut again because the same feelings came back. People usually cut when they feel like they're out of control emotionally, or isolated, from what I've known. The best thing to do is to keep yourself busy, keep your schedule full and try to keep your mind off things. I know it's not as easy as it sounds. It's good that you're doing Karate because at least you have some time out of the house to take your mind off things. Whenever you feel like cutting go and concentrate on something else, go for a walk, read a book, watch telly. Anything.
After I cut the last time I felt guilty and ashamed of myself. I haven't shown them to anyone because the usual reaction to people who cut is "they're attention seeking". This isn't true, it's because they're overwhelmed by emotions or feel alone.
I know you're upset about your Grandma but as they say 'time heals all' and eventually it will stop hurting and you'll be able to remember all the good times

But until then you have to try to contol it.
If you feel like you can't do it by will-power alone then go and see the doctor and they will most likely prescribe anti-depressants or will make an appointment with a pschiatrist (Not the most popular option, but they really are worth their weight in gold).
Don't worry about the scarring for the cuts you have now, use strong salt water and use an antibacterial cream to heal them quickly and, depending on how deep they are, there are options to reduce scarring (bio-oil, Scar Zone, Silicone sheeting). They will never completely go away but they won't be noticable after a year or so.
I hope this helps and I hope you listen to your friends and family, they know what's good for you and they'll always be there for you
Holly x