N.D.
First of all let me give you a {{{{HUG}}}} to one widow from another. It is hard, but you can do it and will do it for your children.
So DD is starting to feel her independence. Well it will get worse before it gets better. She will improve around age 20 or so. When she picks on her brother, remind her that she is older and supposed to show him how to behave not how to be bad. Then if she continues put her in the corner. She WILL hate it, but tell you she doesnt care.
When she 'thinks' she can do things by herself, let her. She will either do it or fail, either way she will learn and so will you.
Be glad she is polite in public and enjoy her struggle for independence. Its hard for 'mother', but really thats what we should want for all our children. And dont worry, you will be mommy soon enough, when she needs you.