Where to find rides to and from doctor's appointments and hospital tests?

Contact your Area Office on Aging or your local Alzheimer's Association. In my city they provide transportation and charge half the fee.

You can find information on similar questions at these threads:

https://www.agingcare.com/search.aspx?searchterm=transportation+to+medical+appointments

https://www.agingcare.com/search.aspx?searchterm=paratransit

If you don't find what you want, click on the 3 white bars to the left of the AgingCare.com name, click on "search" and enter "transportation", "rides to medical appointments" or similar categories and you'll get hits on other questions on the same topic. From there you'll need to search your area in Texas to determine what's available.

I'd start with the local public transit agencies; they're usually the cheapest.

This is a commonly asked question here.