We need two incomes and $10 an hour for me is just not going to cut it. Our other thought is moving back to my hometown where I have free child care with family and taking an office job with full benefits that I do not currently have. My husband can transfer there with is company most likely.
OP your employer's offer to let you bring your child if you reduce from $13:50 to $10 is entirely reasonable. Your employer's request that you agree that you will not curtail your charge's schedules based on your newborn's needs is also entirely reasonable. You can not expect someone to hire you at full salary when you are bringing your child and expect that your charges will be need to follow your child's schedule.
Its good that you are being honest with yourself that you can not see your newborn's schedule not taking priority over the older kids who you think can just entertain themselves. You can't nanny for the family and bring your child with that perspective. Most employers would never consider hiring a nanny who brings her infant because of EXACTLY what you have written.
Very few employers will entertain the idea of a nanny bringing her child. You will face issues if things go south with your employer and you need to find a new job. You are much better off moving to where you can find free childcare with family and switching to an office position where you have access to affordable healthcare. You can always go back to being a nanny once your kids are in school and before/aftercare costs are lower than full childcare.