There are lots of different kinds of kaki and some are very firm, not slimy at all. I didn't use to like kaki for similar reasons, then I tried a couple in Japan that changed my mind. As I said, there are dried kinds too, see if you can find some in the basement of a department store where you can try some before you buy them.
EDIT TO ADD: one of the kinds of fresh kaki that is not slimy or mushy is Jiro kaki (次郎 柿)

The cost of a few nashi is nothing compared to the cost of not eating enough fruit when you are on a diet of onigiri, sushi, makizushi and other white rice dishes.

Just ask at the concierge desk at the IC Yokohama, they'll have a knife you can use, you'll probably have to borrow it from one of the restaurants with their permission, but they will have a knife there.