The state park at Mecham is called Emigrant Springs Campground. Sites are walk-up only. Not reservable. It's a very nice, quiet campground in the trees. Just be aware that if you go there from Pendleton, you get the "fun" of pulling your new trailer up Cabbage Hill. A good way to see how your tow vehicle handles the new TT on a long, steep, winding hill. It's a heck of a grade, but is multi-lane so you can get out of the way of faster traffic and also get around the numerous 18 wheelers that are always crawling their way up the hill.
Grande Hotsprings Resort is a nice park with a pool and WiFi, but you still get the "fun" of pulling Cabbage Hill to get there from Pendleton.
Of course if you are heading to Texas you're probably going to be heading East on I-84 so will ultimately be climbing Cabbage Hill anyway...
Wildhorse Casino RV Park is just outside of Pendleton (basically at the bottom of Cabbage Hill on I-84) and is a full feature RV park with pool and spa. And you can spend any cash you still have in your pockets after purchasing the new TT over at the Casino. Probably win enough money to pay for the TT...
Another fairly close option is the Boardman Marina RV park in Boardman Oregon. It's about 45 minutes West of Pendleton on I-84 and is a very nice park right on the Columbia River.
Personally, if it was me, and I wanted to hang around Pendleton for a few days testing out the TT, and being close to Thompson RV and near the city to purchase necessities, I'd go with the Wildhorse Casino RV park.