I never claimed OP wouldn't have to go to secondary inspection. A few hours in secondary inspection is a minor inconvenience when the alternative is months of unemployment.
The fact of the matter is that there is no entirely safe course of action for our OP. OP will be unemployed in a week and needs to take some action or be unemployed. The I-129 as some have suggested may be the safest route if the employer is willing to go that route. There is no guarantee the employer will go that route. Some employers actually prefer the I-129 since it gives them more control over the process. Others want you to start yesterday and won't wait out the process of a mail in application. We don't have enough information to know OP's situation for sure.
I'm quite aware that I'm advising OP to take a less than ideal course of action. What I'm suggesting wouldn't be my first choice of action. The first choice isn't available, however, due to the combination of difficult circumstances OP is facing: job loss and a criminal charge. Under such circumstances OP may need to settle for a riskier plan, and minimize the risk by preparing a TN package that documents both eligibility for the TN and admissibility to the US. (The safest choice would be to neither change jobs nor travel until the DUI is fully dealt with. That option simply isn't available under the circumstances.)
Yes the denial of admission to GC holders in Trump's first month in office was a serious human rights violation. Even then, though, to the best of my knowledge they weren't detained for weeks or months as you seem to think OP might be. If one wants to minimize the risk of being detained, the best approach would be to fly into the US from a Canadian airport (not enter by land) and have the first US airport on the itinerary not be in the state where the DUI is (for example, if going to CT, maybe landing in EWR and taking ground transportation from there). I agree with I-129 as the best alternative but riskier options may need to be considered if the employer won't go that route.
The advice that others are offering also carries a significant risk: the I-129 could be denied, or the employer not want to go the I-129 route. It could well be impossible to complete the AEP program before one becomes unlawfully present in the US if OP can't get a new TN by mail. I'm not directly familiar with AEP, but I just looked it up, and it seems to involve a lot of steps--not sure if it could be completed in time. Again--we come to the conclusion that OP is in an unfortunate position where there are risks with any course of action--there is no risk-free choice. However, again, I do think I-129 offers the lowest risks if the employer will do it.
I actually have personal experience with all three woes being discussed here: the immigration effects of a relatively minor brush with the law, secondary inspections, and unemployment. For me personally unemployment was, by far, the greatest ordeal. The others weren't even close. I would take significant risk in other areas to avoid being unemployed.