the answer is apply the effect to the whole video. Boom.
It looks like you have come up with the answer to your particular situation. Great.
But, so that your experience can be of benefit to those finding your thread
a. What version of Premiere Elements are you using and on what computer operating system is it running?
b. How were you not applying the video stabilizer to the whole video clip in the first place? Or, are you suggesting
that others apply the video stabilizer to all the clips on the Timeline instead of one video clip that may really need that effect?