Having tried CAD in second year uni I can tell you it's not easy to do or be good at it, so if you have an aptitude that's an advantage.
You have the right attitude, I agree with taking what's available but don't sell yourself short, keep applying, improve your cover letters, resume, and interview technique. One avenue not open to me that is open to you at this stage is the use of the web and online networking through linkedin. I'd make it a point to attend events in the sector you want a job in and meet more people. I know it sounds daunting now, but it will get better with the more people you meet. This will build confidence and I think you need some to keep you forging ahead. Linkedin is populated with HR executives and HR Managers. I know you want to get to the Technical Director and put yourself in front of them but you need any avenue you can get.
Remember mechanical is really at the backbone of anything that is manufacture, that moves, pretty much anything people take for granted. Automotive, Aviation, Energy, Electronics, Building and Construction all value a solid mechanical engineer.
When I landed my job in Singapore back in 2001 I already had working experience in Australia. My first job in Australia came after a "lucky" break and landed a role in applications for an instrument company. My job in Singapore took some 50 applications in two weeks, I moved here, stayed, applied for jobs and cold called maybe over 120 companies, basically saying I'm engineering qualified, i'm eager to work, i'm here for two weeks and can you meet me?
Have your application on job websites from that country.
Is there a society or group dedicated to CAD engineers? maybe they have a link to job portals.
I don't see an issue to start applying for Houston jobs then plan a trip. But I don't know the employment status timeline for a non-US citizen seeking to work in the USA.
Has anyone seriously looked at your CV?