I've been on warfarin for quite a few years. In that time, I have cut myself several times just doing the normal things one might be doing in their shop. Oh sure, I bleed more than if I was not on a thinner but it is not tragic. All of my cuts have been handled by applying pressure with whatever is at hand. Usually a piece of folded paper towel held in place by some duct tape for a while will do the job. Really, it's not the bug-a-boo that you think. Now, a really serious deep cut may well require a trip to the ER but you woluld be going to the ER for that kind of cut even if you weren't on a blood thinner. Just use caution and don't live in fear 'cause it just ain't that big of a deal.
Where your big fear should be is when it comes to striking your head such as if something heavy falls on your head or you fall and strike your head. Now, that is serious. When you are on a blood thinner and you strike your head forceably such as in a fall, you MUST IMMEDIATELY go to the ER, let them know what has happened and that you are on a blood thinner. It is possible to have a brain bleed that can do serious damage or even death of not treated properly. They will do a CAT scan of your head as soon as you arrive, probably keep you over night and repeat the CAT scan in the morning to be sure that somethign hasn't shown up over night. If all is okay, they will send you home to resume normal activities.
Blood thinners are carefully controlled by your doctor and just because you are on them, doesn't make you into a hemophylliac. But, always practice safety when using sharp instruments. It's just a good safety practice for all of us. If using a protective cut resistant glove when using sharp tools makes you feel better, do it. Heck, it ain't a bad idea for all of us.