Hyphenated pretty much hit the nail on the head. You can't absolutely trust the barrel markings to determine if yours is a Texan. If the stock serial number matches the frame serial number (1980) and it has an 18.5" barrel it is most likely an original Texan. The "Texan" reference seems to be a semi-official model name depending on the year. As an example, a lot of rifles with barrels marked RC are real Texans. That being said, my 1984 model with straight grip, cross bolt safety, and 18.5" barrel is marked exactly as a Texan of that year with "336TS".
Here's some pretty darn accurate descriptions of Texans in the category yours is in:
Years of production: 1951 - 1984
Basic specs: 30-30 Win & 35 Rem, 20" barrel, straight grip butt stock
Some exceptions to the above standard spec description:
1964 - 35 Rem caliber no longer offered
1965 - Also chambered in 44 Rem Mag
1965 - Saddle ring added to left side of receiver
1967 - 44 Rem mag no longer offered
1970 - Round lever dropped, square lever introduced
1972 - Saddle ring dropped
1979 - Glenfield 30GT offered but with 18.5" barrel
1980 - 336T barrel changed from 20" to 18 1/2"
1984 - First year of cross bolt safety (TS)
1988 - Last year of production of Texan
Some later models were called Texans, designated LTS, and all made with 16.25" barrels from the mid-60's Marauders.
I can by no means testify under oath all this is true but it seems to be the set of information most Texan "experts" can agree on.