I don't know of any program that will give you a free caregiver for how many days you need for a trip. I see from your profile that your Mom has Alzheimer's/Demetria... if there are numerous things she cannot do for herself, you probably would need to all an agency who have caregivers familiar with memory issues. Would your Mom even allow a stranger in the house?
Is is your Mom on Medicaid? If yes, check with your State Medicaid office to see if they have a program that could help out.
I don't know what state you are in, but I spoke once with a lady who was the director of a Senior Center in the state of NC. She told me that they provided respite care for families' who care for dementia patients in their home. I didn't get the details, but I think it was free. I'm not sure how long you got the respite care. It might have been only for a weekend, but I might call your local Senior Center and see if they do something similar. If they don't, ask for a referral. Sometimes just asking a lot of people for referrals will eventually hook you up with someone who might can help you.
Is is your Mom on Medicaid? If yes, check with your State Medicaid office to see if they have a program that could help out.