Is there anyone in the Pacific Northwest who has had those weeds in
their yard that look like dandelions, and succesfully gotten rid of
them for good, and if so, how?

I suspect what you have may be Hieracium spp., hawkweeds. Do they have
long stems on the "flowers"? If so, see:
http://
http://www.for.gov.bc.ca/hfp/publica..._R3-June06.pdf

Easiest control in small areas seems to be to pull them as soon as the
rains start, and reseed with good quality lawn seed.