Who do I call to find a ride for medical appointments or grocery shopping?

If you have resources such as an I-phone to be able to sign up for an app to have an Uber or Lyft account, (and a credit card); it will be more expensive for a ride, and they drop you off. Is technology something you are able to manage?
This may work as a secondary resource for you, the "shared uber rides" are less.
My husband can take Uber 7-8 miles for as little as $3.50 on a shared ride during non-busy times. Other times, non-shared, $10-15.
We try to use many forms of transportation, but there are still times I cannot drive him, so he is functioning better with Uber.
Hiring a personal driver from the senior center list of drivers has also helped in the past-
$10-12 an hour, and the person stays with you, may help you shop.

There are some good options; check with your local transit company to see if they have paratransit services. In my area, it's very cheap (few dollars) but it only serves a radius of 15 miles from your home.

Some senior centers have transit through the local transit agency.

I've read that some senior centers also provide service to grocery stores.

I would start with the county office of aging. They are up to date on all senior services. Most pharmacies and grocery stores now deliver. If you are in the states and UBER or LYFT is near you, you might try them. The state I live in has transportation services for seniors. And, my house of worship offers the services you describe via volunteers.