Note, none of you are actually "changing jobs while under a TN". You are getting brand new
TN visas, which is the law -- your TN is tied to that one employer and only that one employer, period. If you change jobs, you need a new TN. If you try moonlighting and get caught, USCIS will cancel your TN and kick you out of the country.
I think theoretically you even need a new TN if you radically change jobs in the same company, since you have to specify your job, title, and salary in the application. But that may be too much bureaucracy even for the US government.
Note you also can't "renew" a TN visa. Every year you get a brand new one. There is no such thing as renewing.
Fortunately, TN visas are easy to get. All you need is a job that qualifies and a plane ticket to the US. I guess since they are only good for a year, and under a
TN visa you are *not* considered a US resident, USCIS doesn't worry so much about them.
So if you want to change jobs, don't let a TN stop you. However, if you also have a green card application started, that's a whole other story. you don't want to change jobs in that case.