I need caregivers to come into our home to care for my wife for at least 10 hours per week. How do I find a good one?

I can relate.
We called Visiting Angels here in Orlando.
They sent CNA's who were probably getting paid $15 an hour, but we got a $3 work ethic. In the end, I realized I was paying $25 to the agency only to have someone listen to my mom's stories, but not really do the actual work of assisting her? So I stopped their "services." But it can take a toll on you.

So, I spoke to her primacy care physician because my mom is not very cooperative with her care. Luckily, the PCP placed a consult and a nurse came to give the intake and found that mom does need help with medical care and so her insurance will cover most of the cost. Believe me, it's better to go through her PCP. Once the consult is put in, then her insurance company HAS TO follow up on this. And most PCP doctors have some type of coordinator in their clinic/practice that can help you by sending the agencies the consult request, follow up, etc. Ours did.

I would say, take her for her annual check up with her primary care doctor and speak to him/her about this.

AND PS!
Don't ever pay someone under the table for anything. Breaking the law is never a good solution for anyone.

OldDog4321, you may want to check with your tax preparer to see what is the best route to take. I remember years ago my Dad was able to deduct some caregiving cost on his income tax, thus he needed to have proof of such payment

And I remember my Dad's home insurance carrier asking him if the caregiver was from an Agency or not, otherwise he would need to purchase a "workman comp" rider.

My Dad went through an Agency where he would pay the Agency and the Agency would pay the caregiver. The Agency had workman comp on their caregivers, also licensed, bonded, and insured. The Agency had their caregivers get their flu shots, and probably today need to have their covid shots.

I was self-employed [Independent Contractor] but was never paid under the table. I got a check from the employer to which I had a Contract. It was my responsibly to pay my own social security and Medicare, pay my own health insurance, IRA/Roth, etc.

Under the IRS aides are not self-employed. Legit agencies are their employers and do payroll deductions. You are right that if u hire privately ur responsible for the deducts.

10 hrs a week at $20 an hr is only $200. For that much I may pay under the table. Not worth taking taxes out of and maybe hard to find someone willing to work.

The agencies around here (norcal) pay for the CG's insurance, their liability for slip and fall or theft, and take care of their W2s as they are seen as employees, not 1099s. I would insist on that from your agency as well.