That IS annoying. It also seems incredibly time-consuming for the poster, for very little return on investment. (Oh, look at me. all business speak-y!) Perhaps the main purposes for belonging to a Facebook group (and there are many) is to socialize around a given topic or community. It's a major faux pas to use a group in order to frequently and unabashedly promote yourself. Not everyone knows or adheres to this, however.
There are a few things you can do, depending on how passive-aggressive you want to be. The least so is to send this gal a message telling her that her overzealous posting is having the opposite intended effect. You can tell her that if she keeps doing it then you'll be forced to contact the group's administrator and possibly ban her from the group.
Or, you can tell her in a way I think is more effective: the kill 'em with kindness method. You can say something like, "Hey Maria von Spamalot, I've noticed that you're trying to get people to read your blogs. Totally know all about that as a blogger myself. Let me tell you, though, that my experience (and advice from professional consultants from across teh interwebz) dictate that it's not effective to post the same thing to multiple groups every few days. Here are some ways I've gone about it that I've found to be useful ..."
If you're not feeling charitable (and I totally don't blame you if you're not), then just block the offending party, or report her to the group admin, and move on with your life. But I do feel like novice bloggers sometimes really just don't know any better and can benefit from some coaching/Real Talk. We have so much to learn from each other, and I think if we can get passed our initial feelings of "I wish the Twitter Fail Whale would just stomp you to death already," then we can start meaningful conversations about how all these mediums and platforms are changing the ways we connect with people and blah blah kumbaya, you know?
In terms of advertising, the consensus I've heard (and agree with) is that most ads are worthless and ugly, but that affiliate link ads can be beneficial. Here's a primer on that. And I still think that the most meaningful way to gain followers is to post about yourself LESS and other people MORE. More on that, here.