not need to quit before get TN

I never heard of quiting the current job before obtaining a TN1 transfer. Although you are not allowed to work for current employer after obtaining the new TN1, you don't need to quit.
You need to prepare all the paperworks for TN1 transfer, book a flight to Canadian border on a weekend to have TN1 transferred. You only need two to three hours to process at immigration office, so fly in the morning, and return in the afternoon. Once you got the new TN1, you can start at any time, don't get paid from the current one any more.

I have done quite a few times of TN1 trasferred at Vancouvcer border. Last year alone three times. Recently I have renewed my TN1 without atterney via mail, approved within 6 weeks.

Vancouver border seems pretty easy, always remember to ask for the stamp of "multiple entries". I understand that once you are on TN1, changing employer should be aprroved, assuming the positions are similar.