The TX Dth Idx thru 1976 is free on the Family Search site. Have you used it? I only use FREE sites. I've got the Index bookmarked, just copy and paste: https://familysearch.org/s/collection/show#uri=http://search-api:8080/searchapi/search/collection/1320964&hash=Mrd8SMocDIIen2Q83tu%252B82PRagg%253D
You can view the index listed as 1890 - 1976, but keep in mind TX didn't officially begin recording vital records until 1903, and some counties didn't get on board until later. Along with this index you can view digitized copies of the original death certificates and make copies, if you wnat. You will have to register, but no problems with that. Just make it something easy to remember because you have to use the User Name and Password each time you open this index.
One of my favortie searches is using surname of father and mother. Click on parents for this option. Some times you'll find a young child that died which the family knows nothing about. And, great way to find the married daughters.
There is another TX index that goes up to 2000, but no images with that index.
Family Search is the LDS/Mormon site -- https://familysearch.org/-- Scroll down and click on United States, and then you'll a list of various records they have been indexed for different states etc. Get familiar with the list, but check back as it is growing.
At the top of that main page, click on Learn and you can find videos with presentations on how to research in U.S. and foreign countries, how to get the most out of census research, land records and on and on. This collection is also growing.
And, you'll want to look at the Wikis, too. Anyone can add information to a Wiki. I found Hamilton Co, TX had some good information, including a link to the Hamilton Co, TX GenWeb site. But no one had added the link the Hamilton Co, TX Gazetteer. It's a site with tons of information on the county, so I added the link.
Have you visited a Family History Center near you? Is there a genealogy society that meets near you? You often can get help from these sources.
It's hot and muggy here in So CA and if I'm not making sense, don't hesitate to ask questions. I'll try to help you as much as I can.