On my recent trip to Dallas I visited
Fair Park along with
Semantic Compositions, our SOs, and another friend. We went to the park hoping to visit the gardens there, but upon arriving we found that the gardens were closed. My SO quickly discovered that the nearby aquarium was open, but Semantic Compositions and Mrs. SC nixed that idea (darn non-biologists not appreciating nature), so we looked for the nearest directory.
On that directory we found a list of "Special Interest Points":

Since none of us had any clue what a Texas Woofus was, this meant it had to be something cool and thus we set out to find it. First up was to find it on the map:

Of course the Woofus was across the park from where we were, but we were sure the Woofus would be well work the trek. On the way to the Woofus we passed the Cotton Bowl stadium, what looked like an abandoned amusement park, and the portion of Fair Park every tourist longs to see: the State Fair of Texas Administrative Offices.
When we arrived at the location specified on the map all we found was a very plain-looking building labeled "swine".

We figured that most Texans would know to call a swine building a swine building, so we kept looking. We found another map, confirmed we were in the right place, and then walked around the swine building again, but there was still no Woofus. We eventually gave up and headed off to the
Texas Hall of State, and SC and I swore to discover later what a Woofus was.
This morning I'm proud to announce that I finally discovered what the Texas Woofus is: it's a statue that we walked past at least two times (so much for our scientific powers of observation). In fact it is barely visible on the left side of the picture above:

The
Fair Park art and architecture fact sheet (link is a PDF) has more information:
"The Woofus, a combination of six classic Texas animals, sits outside the Swine Building atop a 16-foolt [sic] pedestal and was originally created for the 1936 Texas Centennial Exposition by sculptor Lawrence Tenney Stevens but disappeared shortly thereafter. It has a sheep's head, a stallion's neck, a hog's body, turkey feathers and duck wings. The "crowning" glory of the piece is a set of ten-foot wide Texas longhorns gilded with chromium with a crown perched between them and a Mexican blanket draped around his shoulders. The Woofus was returned home in September of 2003 thanks to the efforts of Friends of Fair Park and the State Fair of Texas."
The mystery of what a Texas Woofus is can now be laid to rest. Check this site in the future for more equally compelling pieces of detective work from Radagast, Semantic Compositions, and team!