Do I have to have a Texas Driver's License to have Texas car insurance?

IF you find a State specific insurance carrier, yes.  If you find a carrier that writes in every State, no.  I have a winter home and a summer home.  I insure my vehicle under the winter address which is where I am licensed.

Ask a Texas insurance company.

You do.  If you don't live in Texas, you can't get Texas car insurance.  And if you do live in Texas, you need a Texas driver's licence.  In order to be valid, your licence has to be issued by the jurisdiction in which you live.

Once you move to Texas, your out-of-state driver's licence is valid for __ days.  I left the number blank because it varies by jurisdiction and I don't feel like looking it up for Texas.  But once that deadline has passed, your out-of-state licence is no longer valid.  There might be an exception or two due to the current pandemic, but don't expect a free pass forever.

You might, maybe possibly, be able to get a Texas auto policy with an out-of-state licence, on the condition that you'll transfer it to a Texas licence as soon as possible.  If that happens, and I said if, you can expect your Texas insurance provider to call you several times a day and ask if you got the transfer done.