The heavily populated areas of Texas---Dallas, Houston, San Antonio, Austin, etc.---would go down in the first outbreak, although they'd put up a valiant fight with their technology and research centers. The suburbs dependent on those cities would go next. A number of ranches near the border, where desperate, infected people would come in the hope of receiving American emergency care, would also be eradicated. At the end of the day, it'd look something like this:
Society would consist of loosely connected homesteads, with very little technology save that what could be salvaged by the occasional risky foray into formerly populated areas. A formal government would be non-existent, and daily life would be governed according to an ill-defined semi-tribal system, with day-to-day survival the primary concern.
Trolls/giants/et.al. do not like sunlight: they are sensitive to cold, and I think it's safe to assume that they are sensitive to extreme heat as well. The "danger times" would be in the spring and fall, when it is frequently overcast and the temperatures are mild. Winter and summer however, would be relatively safe. In order to protect themselves, I think it likely that the survivors would adopt the old Native American technique of setting wildfires in order to drive out the trolls and prevent trees and suchlike from taking root. This periodic arson would be nowhere near as professional as that of the Swedes: the fires would be set, and then the survivors would focus on keeping the flames away from their homesteads. It would settle into a yearly routine---hunting, scavenging, and harvest during the summer, fires during the fall, hunting during the winter, fires during the spring, and repeat.
Cats, including cougars and bobcats, would be immune: so would possums and armadillos which would quickly become meat staples. Other mammals would be threats, not only infected ones, but healthy predators (feral dogs, coyotes) and competitors for limited food sources (raccoon, deer). Cold-blooded creatures would also be eaten.
Immunity would be a very hit-and-miss thing with only the most basic medical understanding and equipment available. The rules for everyone would be: don't ever touch a troll. Wash often. If somebody starts to show a rash, expel him or her, and in two weeks, if healed, he or she can return.
All in all, the Texan survivors would have it pretty tough: I expect life expectancy would only be about 20-40 years, and that there would be a high infant/child mortality rate, for which reasons large families would become a necessity again. Life would be hard and short. But they'd survive as best they could, hoping for the day when things would become easier.
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