I won't lie, it is irritating to see someone deliberately step around the rules. Yes, the immigration process is long and expensive. I know - I've been through it. Yes, there are a lot of hoops to get through, but there are websites where you can keep up to date on correct policy/procedure for the process (such as UK Yankee).
I think you likely know that getting married on a visitor visa is a no-no, and you went downhill from there in following rules. I'm baffled as you say you've flown back and forth, but it sounds like you've been here ever since you got married. I find it hard to believe that immigration didn't peg over 5 years that you were circumventing their rules. The dates in your passport would have given you away immediately. I suspect what you really did was get married and then overstayed for five years. Now you're looking at DLA and carer's allowance and things like this, and you've realised that you need to be here legally for things to be smoother for you to claim. Were your dcs born in the UK? you wouldn't have been eligible for free NHS care, so I'm curious how you bypassed all that, as I doubt you went to the US to give birth. And have your dcs even been registered as US born abroad? Likely not, as I imagine it would have drawn attention to your own immigration status.
Sorry if I come across as unbelievably cynical, but your petition is tellingly vague in saying what your actual actions were.
I'm not saying that you're a troll, I imagine you ARE in a difficult position at the moment. But I think you're being sparing with the facts, as it would then be pretty clear that you attempted to bypass immigration controls and got caught in a bind because of it. And that's really frustrating to those of us that jumped through all the hoops, no matter how expensive and frustrating and time consuming.
I hope, for your children's sake, that you get to stay with them, but I would also hope that you realise that this situation was of your own making and could have been avoided.