A.R.
Your legal name is attached to your SS#, but you can go by any name you want and usually it presents no problem. For example, I have bank accounts, drivers license, etc all in my maiden name, but my SS# has my married name. I only have to use my married name when filing taxes or other things government related.
You can pay to have your name changed with the SS but I can't remember the cost. Also, it is granted at the divorce that you can change your name (and I believe at that time it is free to do so with SS if you have that document from court.)
For your children, the father must approve it unless you can get other permission from court... or at least that was the case in NC and in every case I have heard of elsewhere. IL might be differnt, but I doubt it!