小草,you know the best about your hu*****and, about whether or not he is a responsible person and why he wants so badly to take this offer. I also believe that you have actually made up your mind about supporting your hu*****and but you would stay where you are now to provide a stable environment for your kids. But you are not sure whether or not you are mentally and physically strong enough to take care the kids while your hu*****and is far away and that's why you are seeking help here. You can! If you believe yourself, and trust your hu*****and.
My hu*****and's career path was in the similar pattern years ago (8 jobs in 8 years), and what not similar to your situation was that I stayed at the same place and the same job for all those years, until we finanlly settled down where my hu*****and is now. He is very happy about his current career and has reached a level he never dreamed before.
If you haven't decided to move or not, try to negociate the relocation package. If it is a fixed sum, there probably aren't much you can do. But if it's a reimbursement of the real cost within a period, try to extend the time limitation up to 2 years, which will give you more time to consider relocation.