Another FB question

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DH has a cousin who we see maybe 3 times a year. She discovered facebook about 6 months ago. This cousin is in her thirties, extremely introverted, doesn't work and she is on facebook all. day. long. Cousin friended DH and I on facebook and we accepted.
Cousin will message DH and I at least once a day asking general "how are you", "what are you doing today" type questions. I don't check my facebook everyday, so I get back to her when I get back to her. However, lately it's become a problem as she gets upset when DH and I don't talk back to her immediately on Facebook....to the point where DH's aunt will call and talk to us about it.
A couple weeks ago I posted something about visiting DH for lunch and then going out with friends...cousin posted 4 times on my status asking how it was, what did we do, etc. I didn't respond because I had some things to do. I later posted something else on my page and Cousin private message me asking why i didn't respond to her and she saw I had posted something else so she knew I was online.

Other people outside the family that are friends with Cousin have said things to DH about it and they have defriended her.
Normally, I'd just change my privacy settings or defriend the person, but I feel there's sticky issues because she's DH's family and because she's extremely introverted. I know she'd notice that the settings have changed since she's on there constantly.
I dont' want to hurt her, but she doesn't seem to take the hint that we're not on there all day or won't respond right away, if at all. We aren't close and she doesn't talk to us in social situations, just on Facebook. I'm not really sure what is the polite way to handle this.