I figured out what is wrong with the 375. Nothing is worn out or broken – it seems to be just sticky from not being used for decades. See the pics below. The piece that rotates and catches the firing pin sticks down. I put some Kroil on it & it loosened a bit, but not as much as expected. I can rotate it up and down by hand, but it’s still stiff. Maybe it’ll be looser tomorrow.
The Dumoulin trigger assembly: You can see the silver colored sear projecting from the right side of the housing.
This part pushes down, but doesn't come back up. That's what keeps the firing pin from falling when the bolt is closed. Usually Kroil loosens things like that in a hurry, but not yet on this.
This is the other end of the same part. It pivots on the pin on the left side of it, and is engaged by the sear on the right. If I push down about where the magenta line is pointing, the other end of it (in the previous picture) comes back up. Safety and trigger then work for one shot, then it sticks down again.
I believe this rifle is sold as-is anyway, but I really wanted to find and fix the problem myself, just for my own knowledge and satisfaction. Tomorrow, I should have a firm commitment from the buyer. Same buyer also badly wants the 416 and has offered me a LOT more money than I'd already agreed to sell it for, but I won't break my word to sell it to the first buyer.
BTW, I'm the same guy who was selling a Heym double in 458 Win Mag last February or so, and you folks here were great help, and a member here ended up buying it.