Can anyone provide a list of companies in Thurston and Pierce Counties
of Washington State; which have positions in Material Handling,
Warehousing, Forklift Operation, or similar positions that start at
$12 or more per hour for a person with more than 10 years of
experience?

I don't kow what the pay for these jobs is, but if you haven't done so
already, I suggest going to the following page:

http://www.wa.gov/esd/work/advancedjobsearch.htm

Click the boxes for Tacoma and Olympia, then put "forklift warehouse
warehousing material handling" (without the quote marks) under
Keywords, select "ANY Words,"  and press Submit.  There aren't a lot
of relevant jobs listed there, but there are some.

Best wishes!