I'm on blood thinners, too, but still use a Gillette DE safety razor that is a year older than I am. I just pay close attention to my shaving and keep a styptic pencil close by, in case I do cut myself. I just use ivory soap to make a lather, no fancy stuff for me. Use 91% rubbing alcohol for aftershave.
I started shaving my head, last week, but have not started using the DE razor for that, yet.
I started shaving my head, last week, but have not started using the DE razor for that, yet.