It's certainly not well-publicized, but there is a growing collection of old newspapers online thanks, in part, to the University of North Texas.
I got curious about the availability of way-back issues of Texas newspapers after reading about the online-archiving project undertaken by the state of Indiana. With assistance from the National Endowment for the Humanities and the Library of Congress, Indiana's state library is putting fully readable copies of very old newspapers online.
I wondered whether the Texas state library and archives commission was doing anything like that. Turns out, they aren't. Instead, UNT is the lead organization in the state to digitize old papers with NEH and LoC aid. UNT's website says it has other partners.
Seems to me (and, yes, I have a vested interested as an old newspaper guy) this would be worthy of state funding.
By the way, right now only a relative handful of Texas newspapers have way-back issues online as part of the UNT program.