Well, premium processing is not absolutely necessary, since all that is required to keep your TN status is filing the I-129 by April, not necessarily the approval. I merely suggested premium processing to get the TN pushed back to you faster so that hurdle is past. You don't want the pending TN laying around CIS office while you file all your GC paperwork. This might get you a TN denial, which would make you have to stop work until EAD is awarded.
Marriage to a USC who lives in US doesn't quite prevent your from getting a new TN (that is why I've told you to file I-129 for TN -- you would get it -- and then file for GC), but it does presume you to have immig intent, which would disallow you to enter US on an existing TN or getting a new one at the border (that is why going to the border is risky). However, if you just go by yourself, no wedding ring, and don't announce that you are married, you should be fine. But any trip outside US at this point is risky -- one of the reasons we always remind TNers who marry USC to marry, honeymoon, and stay put in US until EAD and AP are awarded.