Since they are the ones who need to file the form requesting your PERM, they would be your “sponsor”. If a bachelor’s degree is required for your certification, that helps. Either way, they would need to obtain a Labor Certification from the Dept of Labor that basically says there are not enough workers with your necessary training, skills, education, certification, and employing you will not effect salaries of Americans in your profession. Like if you were working for minimum wage, but Americans usually get paid 10 times that or something, they wouldn’t allow it.
The thing is, it would be your employer (the sponsor) who is petitioning. You can’t petition on your own behalf unless you discovered cold fusion or something and wanted to continue developing it in the US.
One good thing about the PERM request, is that while you are trying to get your Labor Certification, your eligibilty to renew your TN Visa is not effected. So if your Labor Cert is denied or while it is pending, you can still renew your VISA and you can still travel back and forth while it is pending. But once it is approved and your employer goes ahead with the PERM petition, you have shown your intent to immigrate.
So, just like filing for the residency through marriage, you would want to initiate it right after you renew your TN so that you have the maximum amount of leeway and flexibility. You might need that entire year for all your stuff to be processed and get finalized. You might run out of time though, and you’d be screwed because you wouldn’t be able to renew the TN.
Since you have the Bachelor’s Degree, your employer could sponsor you on a H-1B Visa. The process is similar to the PERM but it is a nonimmigrant Visa, so it wouldn’t effect your TN Visa. There are limited ones issued every year, but they are good for like 5 years. Much longer than your TN Visa. That would give you pleanty of time to have some romance, get married, file your I-130 petition and immigrate with your new wife sponsoring you.
That might all be irrelevant since your employer doesn’t sponsor people.
So… find yourself a nice nonprofit organization that is in need of your special medical tech skills and is willing to sponsor you under an H1-B Visa. (The annual limit to those Visas does not apply to nonprofits) Then, while working under that Visa, get married and submit your I-130 for permanent residence. You will have pleanty of time under that Visa.
I believe you can hold both the TN and H1-B. Applying for or renewing one shouldn’t effect the other because they are both nonimmigrant. So start looking around for an H1-B sponsor. You have a good chance with your education, certification, experience and profession. That’s my nonofficial 2 cents anyway.