Now you are down to the wood and repeated applications of linseed oil or its derivatives will make the stock OK, just not original.
There are two distictly different paths to take as far as a finish goes.
#1- would be the traditional oil - Linspeed (speed ), Tung or Danish - or any variation.
#2 - a surface coating, such as a varnish, poly or epoxy (or a converted (two part) urethane - or one of the newer 100% acrylic lacquers".
The better finishes - the two part ones, in catagory 2 offer the best over all protection, but, if and when they get banged up, they are miserable to strip off and redo.
The main advantage of the category #1 finishes are - a quick wipedown restores them to like new looking. The downside is - they suck as far as protection goes.
If the shotgun is going to see any field use in wet weather, this type of finish is going to need a lot of upkeep.
The mid strength - surface type finishes, such as varnish, poly, oil/varnish blends - are the most common. They offer some protection, but, they come off pretty easy if/when they get dinged up.
Shellec is also an option.
The old Soviet bloc AK's all had a shellac finish on the stocks.
It's ugly but utilitarian & subject to discoloration (white cloudiness) if it gets wet or comes in contact with anything that has ammonia in it.
One quick note - whatever you choose.
Only apply oil or finish to the outside - the parts you can see.
Don't finish the insides - such as the forend or the butt if a plate is installed.
Oil especially. It can cause the wood to swell and crack.
Good luck & be sure to take some before and after pictures!
Oh yeah - you can try some Goo-gone to see if it will work on the crud that's still on the stock.
Personally, I'd probably just start sanding it & use some Citrus remover in the checkering - if there is any - and start from scratch.
I've done a half dozen or so in the past. It's sort of fun to bring something back from the dead - so to speak.