I couldn't find a breakdown of registration by party for Harris County. Maybe they don't do it that way in Texas.
There is no party registration. You can only vote in one primary (they might even be in separate buildings, or at least rooms). You are considered to affiliated for the rest of the election year, but that in practice means that you can't switch parties for the runoff. Officially, everyone is cleansed of their partisan alignment on January 1 of the odd-numbered years.
In 2016, Republican voters outnumbered Democratic voters 3:2 in Harrison County.
Trump, Clinton, and Sanders are from New York City (Youtube)
And who do you think got most of those votes? It certainly couldn't have been native Texan Ted Cruz...