The safest way for you, if they are filing I-129, is to ask the new employer, as a courtesy to you and your current employer, to request option "d: New concurrent employment". Then, when you get your second TN, you can transition more smoothly to your new job. (Remember that notice of leaving is never mandatory). You could mention that you are sure that they would appreciate the same courtesy if you were leaving them.
Otherwise, as you suspect, the moment the I-129 is received by your new employer, your old TN is invalid for work, with no notice.
If for some reason they insist on using "Change of Employer" (usually an uninformed lawyer is involved), the best you will be able to do is monitor the USCIS website for the status of your petition, and notify your old employer as soon as you know the new TN has been approved. That would give you a few days while it is being delivered.
I trust they are doing premium processing?
The simplest way, would be to go to the border, get a second TN (keeping the first).