Is it proper to say "Texan accent" or "Texas accent" - or does it make any difference?
I've never heard anyone say Texan accent, while Texas accent is pretty common. Ngrams agrees with me.
On the other hand, you won't hear anyone say England accent or America accent over English accent or American accent. I am not sure about the reason behind this inconsistency but I suspect that perhaps with non-countries, the region/state/city name is used, regardless of the existence of an adjective form.
One speaks with a Texas accent, but one talks Texan. See this article.
EDIT: By request, a relevant excerpt from the linked article. Note the distinction made between a "Texas accent" and "talking Texan."
The Texas accent "has great symbolic value. It has a local identity versus, say, Arizona English. That makes Texas English more resilient," said Lars Hinrichs, an English language and linguistics professor at UT Austin.
There are many aspects to "talking Texan": pronunciation, cadence, syntax, not to mention vocabulary. And, technically, there are several Texas accents — the drawl of East Texans like Matthew McConaughey, say, or the nasal West Texas twang of Laura Bush.