Sorry to hear about your heart problem.
I've also been on blood thinners for the past 25 years or so, so I can relate to your concerns.
Cuts or bruising of any sort can get really messy, but you learn how to live with it.
These days I shave my head daily with a razor, (HeadBlade ATX), have been doing so for over a month now and only experienced one small nick, and that was on a mole.
Believe me the clean smooth feeling of a well shaved dome more then compensates for the worries of a nick or 2, just be careful, and proceed lightly.