Committed but not trusting

I'm still reading in the Old Testament. Yesterday and today King Asa of Judah has been in my reading. I'm catching a glimpse of something I've not noticed before and I need to get my mind and heart around it.
What is one goal of many Christians? To be fully committed to God, right? I would say that has been one of my goals. In fact, I think that if I felt I was, I would have 'arrived'. But I think there is another level, as seen in Asa's life, that can be missing, even when I am committed.
2 Chronicles 15:17 says that Asa remained fully committed to God throughout his life. That's awesome. One would think that meant he lived his life fully as God wanted him to, that he lived in victory.
But in chapter 16 it tells how Asa didn't trust God to protect him against King Baasha, but turned to the King of Aram, even though he had seen God's protection against the Cushites. And so, because he had turned to a man for protection, instead of to God, he was at war for the rest of his life. And even in his old age, he refused to go to God for healing when he was sick. he chose instead to consult physicians, but never sought the Lord's help.
I don't want to make the mistake of applying this with too broad a brush, but I think there is a principle here that is important.
I can be committed to God without trusting God. I can try to fix things on my own or go to someone other than God for help. When I turn to someone or something that is really not all-powerful for help my help is temporary and short lived. It's only when I go to the One who can provide complete victory that I will experience true victory. So if I have areas of conflict in my life, I need to ask myself, what or who am I trusting in those areas.
Of course, even when Asa was trusting God, there was conflict in his life. He didn't live in continual peace because of his trust, but God gave him victory at the end of the conflict. And God wanted to continue to do that.
Now my question is about how this verse fits in. "The eyes of the Lord search the whole earth in order to strengthen those whose hearts are fully committed to him." If Asa was fully committed throughout his life, why did this not apply to him?
Anyway, this is one of those things I need to mull over and figure out how to apply. There are areas where I have continual struggles. Is that because I am not fully trusting God? Perhaps it is so. If so, how can I move into full trust? That's where I want to be, dependent on God, not on my own, or anyone else's resources. I just need to figure out how to get there.