It might help to have more context about how you care.
My first take without that is to recast your statement like this:
"I care about getting things right and/or doing the right thing but sometimes it leads to me losing sight of the bigger picture."
Then next thing they might ask is for an example of when this has happened. It's certainly one of the questions I've asked in interviews (I've done several hundred interviews over the years), and you will be asked about your greatest strength too if they are any good at interviewing.
I have to disagree with the person you paraphrase above. I think the most important questions to have nailed are "Why are you the right person for this position?" and "Why should we hire you over any other applicant?". Those questions are about focusing on your *strengths* and I've found them more convincing than answers to negatively focused questions.
Most importantly, remember to breathe and be as relaxed and loose as you can be while being alert and present. Everyone expects people to be nervous at job interviews so no one is going to worry too much about a little nerves.