Your story is very similar to mine, so maybe you'll find some useful thoughts on my blog: http://www.polyamory.com/forum/showthread.php?t=4540
I also highly recommend this site, especially this section: http://www.morethantwo.com/coupledating.html
How long has it been since you and the woman became involved? If this is a relatively recent development I can see why she's afraid to let things get too attached. That said, I'd be worried if things stayed like that indefinitely. Feelings happens and you need to be allowed to feel how you feel. If it's been a while, and you really feel stifled, I might try explaining to her, "Hey, I'm an emotional person, and I really care about you and your family. Sometimes I I need to express that in order to feel comfortable and to let go of my worries. It's not about me trying to make our relationship into something it's not, I no and accept that we have limited time and resources for each other. I don't want to make you uncomfortable but I hope we can get to a point where you are okay with hearing those kinds of things from me now and then, because right now it seems like it upsets you. Am I right about that? Can you tell me what it is exactly that worries you about it?"
On the other hand, if things are still pretty new, ANY person who's still figuring out how they feel about a new partner might get uneasy hearing gobs of emotional processing and vulnerability. So, I guess it's up to you to figure out where this relationship is and where you want it to go. Have you and she ever had that kind of conversation? Is this a purely casual thing in her mind or could it get more serious with time?
It's certainly possible to develop something serious with someone who's married with children, especially in a situation like yours where you get along well with her spouse and children. For me, it's been a matter of showing that I'm willing to "put in" as much energy as I "take out" from my partner's already overwhelming life. I babysit, I help with projects she's working on, I respect her limited time by making an effort to be part of her life and understanding that it's going to be more rare that she can reciprocate and be a part of mine in the same way. It's not always easy, but I've found that for me, with this woman, it's well worth it.
I would encourage you to open yourself fully to the idea of finding another partner(s). Whether it's casual flings or is something that becomes more serious, it can be very helpful and affirming to be reminded that you deserve and can have more then the limited time and attention you can get from someone who's already partnered. You can do this while being up front about the fact that you're already in a poly relationship with someone else that you're not planning to leave. This may require you to do some educating of your new partners about what poly is, but people can and do accept it, as in the case of my bf.
Good luck, and feel free to ask more questions or seek more feedback!
EDIT: Ah, I looked again and saw that it's only been a month. Sorry, for not reading more carefully. Yeah, that might be a little soon in any relationship in the getting too emotional/attached. But you do have the right to know what she thinks might be possible down the line.