Dad is a WWII veteran, has 24-hour health aides with 2 working 12-hour shifts. Will the VA pay for one shift if Medicaid pays for the other?

It might depend on his rating, somewhat. Benefits office should be able to help.

SonOfVet, it's certainly worth a call to either the VA or to your state veterans service office. The maximum VA Aid and Attendance (A&A) benefit for a single person is $1,830 per month, so it's not going to cover the full cost of a 12-hour shift for 30 days per month, but it would put a dent in that cost. I just found out yesterday that in my state the A&A benefit is not counted as income by Medicaid, so if you can get A&A for your dad, it might not adversely affect his Medicaid benefit. A&A is based on need, so I think it's worth investigating.

This website says that while it is theoretically possible to collect these two benefits at the same time, it is unusual.

https://www.payingforseniorcare.com/medicaid/veterans-aid-attendance.html