Here's what Cruz said on H-1B when in the Senate.
There is currently a serious shortage of workers in the fields of science, technology, engineering, and math, yet every year we send thousands of high-tech graduate students back to their home countries to start businesses and create jobs. This makes no sense.
Donald Trumps comments during the News Conference when he signed the GOP Pledge, on 9/3, use almost these exact words. I can't find a written transcript of that either, but the videos of it are around. About 40 min long.
For further evidence, Google, or Bing, Cruz, Sessions, and Immigration together, and look at all the times they have been in lockstep on the issue.
Trump, Sessions, and Cruz are united on immigration reform.
Sorry, I usually can provide direct evidence, but I can't find a copy of what I heard in Cruz's speech Q&A that day. If I run across a transcript, I'll get you a link, or a copy.
False: "the country has more than twice as many workers with STEM degrees as there are STEM jobs. Also consistent with other research, we find only modest levels of wage growth for such workers for more than a decade. Both employment and wage data indicate there is no shortage of STEM workers in the United States." - http://cis.org/no-stem-shortage