Moving my Post-Stroke,Nursing Home bound mother from Ohio to Texas

Last August my mom had a stroke at home that was pretty severe. Her wits are pretty much with her though she can be forgetful. Her body though has not recovered. She has been in the nursing home since that time and will continue to need that level of care for the rest of her life (she is 68).

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She didn't really have any money and her house was majorly underwater so she doesn't really have any assets so Medicaid was established in Ohio after Medicare's 100 day period where they paid.

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In Texas, Medicaid rules stipulate that she will require a 30 day period before her Texas residency begins and when she would qualify for Medicaid in Texas (from my understanding of it all)

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My plan was to move her here on June 30th and I will be her home healthcare aide until she earns her Texas residency by the end of July and hopefully move her into a facility I've found here on August 1. I'm a teacher so taking the time from work is a non issue.

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My main concerns are (1) my timeline is tight, if she can't get into the facility on August 1, I'm in trouble because I need to be back August 6. (2) I think Medicare provides for some home healthcare aide time a month but i'm not sure how that works and how I would set that up and get someone here. (3) I don't know what I don't know and I don't want to make any mistakes here. I'm hoping someone out there has had a similar situation and can steer me away from any potential pitfalls.

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Thanks for reading!