I'm certain that you care about your father, KC. And (even though you don't understand why) you're not ready to visit his grave yet. Maybe you found his death very painful & you're having difficulties coming to grips with it? I don't know you nearly well enough to know whether this is the case or not. But it sounds rather like it.
None of this means that you care any less for your father. You'll go when you're ready, you know that. I don't think there's any one acceptable version of experiencing grief. You're doing it the only way you can at the moment.
It also sounds like you'd like to take a break from your designated family role of being "the strong one". I did that in my family & well ... it can be a lonely position when that's expected
of you, all the time. And a hard one, too, when you're feeling particularly vulnerable yourself. Like now, for you, by the sounds of it.
Do you think you might feel a bit better if you had a quiet talk with your mother & told you how you feel?
Another thought: I've never experienced grief counseling myself, but others I know who have, found it very helpful in coming to terms with loss & grief. Just something you might like to consider, OK?