I have tried to fix this persistent drip. When I disassembled the fixture, the retainer screw appeared to be broken -- the head was missing and the remains of the screw at the broken end was crumbly even though underneath the debris the remaining end was shiny copper or brass. Looks more like copper I turned the water on and flushed out the valve seat washer (it had not come out with the stem) but no screw head.
The washer seemed in good shape so I reused it, replaced the screw, also replaced the packing washer at the handle end; put it all back together and turned the water on. No drip -- success I thought!

But, the next day, the drip was back. I just finished replacing the valve seat washer but still it drips--one drop every 8 seconds.

Any suggestions?

The only other thing that I think might be useful information -- I disassembled the other outdoor faucet because of a leak at the handle when on that I couldn't tighten down. Took it all apart just to check the valve seat washer and found that it was covered with (I think) mineral build up such that I couldn't even see the screw. And, I couldn't scrape it off!

This faucet went back together and the handle leak fixed okay. I wonder if there is similar buildup in the first faucet that isn't letting the stem seat properly? if so, how can I fix that?