Crimson, I don't know how old you are, but take it from me, you need to be GLAD that you haven't been in a whole bunch of weddings. They can be time consuming, stressful, and EXPENSIVE. I have been in more than I can count - TOO damn many at this point. I recently made a VOW to not be in any more weddings unless it is one of my sisters or one of my SUPER DUPER TIGHT Big F Friends. And when I get married, it will be me, my two sisters, and maybe two other folks tops.
I agree with Kimmie1913 - folks take this whole bridesmaid thing too seriously. I also agree that you need to answer StraightBOS's question about whether you ACTUALLY FEEL offended or not. If you don't feel offended and want to help your cousin out, then go for it. You sound like you really want to do it.
But me, myself personally, if I wasn't close to the cousin and hadn't been close to her since we were 13, then I wouldn't be in the wedding. But like I said, I've already been in 8 go-jillion weddings, so that's just me.