Are both lasers on an optical table but with the spectrum analyzer on
a separate shelf or cart vibration isolated from the lasers?


Yes and yes!
I am starting to think the frequency jumping I am seeing might be because of current ripple in the PSU.
Looking at Sam's laser FAQ there are at least two options. Either the ripple reducer over he
http://repairfaq.ece.drexel.edu/sam/smhpsrr1.gif
or perhaps just more filtering capacitors on the output like many of the circtuits he (many 450V caps in series, with resistors in parallel over each cap)
http://repairfaq.ece.drexel.edu/sam/laserhsc.htm

would a typical "brick" type HeNe PSU work with something like this on the output?:
https://www.dropbox.com/s/epbwgox0tm...hv_filter2.png


Anders