“Kawaii” seems to be a well known word. I sometimes see “kawai” too. But “kawai” is just a family name in the Japanese language. So you shouldn’t use “kawai” if you want it to mean “cute”.
They are pronounced differently in Japanese. When “kawaii” is pronounced, the last “ii” is extended, whereas the last “i” in kawai is shortly pronounced. (This difference is very important in the Japanese language.)
In Japan, it is quite common to call others by their family names. In my case, nobody calls me by my given name except my family and my relatives. So if someone outside of my family calls me by my given name, it feels like my mother calls me and I’m not used to it.
In business, usually family names are used. Among friends, some are called by their given names and some are called by their family names. It depends on which is easier to call someone by their family name or given name. Of course, some are called by their nicknames.
The title of this entry is “Kawai is kawaii. I’m neither Kawai nor kawaii.”. Maybe you know the meaning now. It means a person whose family name is “Kawai” is cute. I’m not cute and my family name is not “kawai”.
I see some people naming themselves like “KAWAII xxxxx”. As a Japanese person, I have to say that’s strange. I believe they are cute. However, it is important to be humble in Japan. You shouldn’t say good things about yourself too much.