The Woodlander I remember was made in Spain by Sarasqueta(sp?) and imported by Stoeger in the 50 and 60's.
Maybe they have a more modern version out now. There's an 'Uplander' shotgun on the market but I think thats mgf in Brazil.
So many of them around now on the market and they come from all around the globe.

Sears used to sell the Woodlanders and other Spanish made SxS's too with their own Model designation and name on them.

IMHO it's most likely worn sear nose(s) and/or worn sear notches in the hammers if it's the old Woodlander from the 50's/60's.
Light weight sear springs could also be in the mix allowing the problem.

They were pretty good shotguns but the Spanish guns of the period just didn't always have the best steel and heat treatment in their parts. Soft sears and engagement surfaces were common.
They made some spectacular guns as well as some so-so guns.

If the gun 'doubles' while dry firing it,,I might also consider the sear arms are touching each other at the back end. Both being lifted with either trigger being pulled.


Also check closely while dry firing by looking at the second trigger while pulling the other,,see if the second trigger is being lifted/pulled along with the intentional pull of the first.
That would indicate something interfering with the separation of the triggers,,a splinter of wood, a piece of a screw dropped in there during reassembly at some time,,ect..I've seen both.

I'd guess the worst for the gun in that the sear noses and /or the sear notches in the hammers are worn.
New parts for these about non-existent. Any around are those stripped from worn out guns so you get about what you are having the problem with. Nothing is a drop-in fit in these from my experience.

Sears can be reshaped and rehardened. Sear notched can be rebuilt, shaped and hammers rehardened.

It can all be redone and the gun made to work very nicely if this is the problem (which we really don't since we can't see inside it!)

But assuming the worst, it's an expensive repair if you have to pay for it. The shotgun is probably a $400 gun off a gunshow table (Spanish, Double trigger ,extractor gun).
So don't get too awful deep into it.

Just my thoughts.