2 Frazzled,
I missed it that the screws actually pulled out. I thought they loosened themselves to the point of falling out.
I feel the best method for your pulled-out wood screws is to install slightly fatter ones to create new threads into the wood. Fatter wood screws will also hold better than the original ones did before they pulled out.
When locating more holes to add more screws, if your rails are like our rails, you will need to slide the rail in and out to see possibilities through tiny openings of opportunities that come and go as the rail moves along. It is easiest if you separate the rail halves to expose every opportunity at once.
Some wood screws have thin/shallow threads. Other wood screws have deeper aggressive threads. Try to find slightly fatter screws with deeper threads.
Here is a shallow-thread screw.
Here is a deep-thread screw which "bites" deeper into the wood and retains more wood to prevent the screw from stripping out.
