This is a wonderful topic!
The "self" cannot being fully controlled, so therefore it is not "yours".
What is being cherished is the idea of a self at its best. We cherish the idea of what we wish we always could be. We do not cherish the idea of a self at its worst. Self "at its worst" is often not considered self at all. However we cannot always choose which one to be. For something so uncertain and unpredictable, how cannot it be declared as "mine"? This is something good to focus on - You have this idea that you cherish, yet how often does that idea actually come to reality? And if and when it does, why does it not stay forever? According to the Buddha - because it is not under your control and it is not yours.
Cherish the good that you are capable of, just try not to cherish the idea of "you" that is doing it.
I believe that if this advice is reflected often enough than self-cherishing will fade away. Self-cherising, just like clinging to the self in any other fashion, goes away completely when the notion that the self even
exists is truly realized to be false.
When you notice that you are cherishing yourself, in what ways are you doing that? Maybe it's something really specific, like your cherish yourself mostly physically and are involved in a very healthy lifestyle or beautification. Maybe you cherish your altruistic actions and when you are doing good for others that is when you are cherishing yourself. I think it's possible that if you can catch that in the act and probe into it you will have much better control over it. And doing the same thing when you make a 'mistake' and think "oh that wasn't me, I lost myself for a minute there when I yelled/lied/etc." probing into that same belief, seeing all the aggregates clearly. Than not only does it become more clear that the whole combination of things is not you at all, but it becomes much less attractive!