The best collections of Texas newspapers are in the archives listed in the "Archives and Libraries" section of this outline. The Family History Library has microfilm copies of a few newspapers, such as the Galveston Daily News from 1865 to 1900.
Guides to help you locate newspapers are:
Sibley, Marilyn McAdams. Lone Stars and State Gazettes: Texas Newspapers Before the Civil War. College Station, Texas: Texas A&M University Press, 1983. (Family History Library book 976.4 H2smm.)
Texas Newspapers, 1813-1939: A Union List. Houston: San Jacinto Museum of History Association, 1951. (Family History Library book 976.4 B3h; 1941 edition is on film 1036813 item 4.)
Some obituaries and other vital statistics have been abstracted and published. An example is
Pearl Hendricks, comp., Texas Obituaries: Death Notices Taken From Early Texas Newspapers, 1835-1890. Two Volumes in 1 typescript at the Clayton Library in Houston. (Family History Library book 976.4 V4t; film 845442; a revised version with the dates 1832-1885 is on film 1398600 items 6-7.)
Use this listing to find the title and location of an ethnic newspaper in which your Texas emigrant ancestor may have appeared: Texas Newspaper Collection by Ethnic, Religious, Professional or Political Orientation (Center for American History at the University of Texas at Austin) http://www.cah.utexas.edu/services/finding_items/newspapers_ethnic.php