You don't attach them to the dry wall panels.
You attach them to the studs behind the panels.
Drywall has next to zero holding power.
Use finish nails long enough to go through the molding, drywall and at
least a half inch into the stud.
Pre-drill if you can't drive the nail with a hammer. With oak, I think
pre-drilling is a good idea.
Use a nail set to drive the finish nail down into the oak and then
either caulk or wood fill.
Someone else said, the problem with adhesive is removal.
If it's construction adhesive, the paper will come off the drywall.
If I had enough nailing to do, I'd use the opportunity to buy a
nailer, I love new tools.
But I can do a decent job with simple tools when I have to.
You'll likely have to have pilot holes.
You need nail long enough to go thru moulding and drywall thickness into
the supporting stud by about 3/4" or so.
Or, for something that light you could (altho I don't really recommend
it) use adhesive instead and not worry about where studs are much.
We're talking simple quarter round here. If it is not at floor level
where it could be contacted with vacuum cleaner, stick it on with
construction adhesive. If it fills the space where the flooring meets
the base molding, rent a nailer and have done with it. Drilling and
nailing he old fashioned way will look amateurish and leave scars
every place you mess up. And you will leave dings and burrs.
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