Clean the checkering with a stiff tooth brush or similar.
Try a little linseed oil. Slap it on a piece of bare wood, don't bother to sand or other prep. What you'll find is that it won't cure quickly and will remain sticky for months.
Besides that, linseed oil is not waterproof.
There are no end of people that disagree with me about linseed oil. That's the reason I recommend everyone use it to finish a rifle to get it out of their system.
Folks still slather it on stocks because that's how those grandpa Marines in the last century treated their rifles - one coat a day for a week, one coat every week for a month, and one coat every month for a year.
Its poor qualities led to finishes such as Linspeed and Birchwood Casey Tru Oil modified linseed based oils that work well. The trouble is the time required to wait on each cost to cure.
A linseed coated stock can be cured quicker by putting it in direct Sun for several days until it's less sticky. They look great on a properly prepared stock after it dries, but too much time and hand rubbing is needed, especially for a gun that will be used.
You should think about staining the stock and filling the grain. The best way to fill the grain is by light wet sanding with a thinned oil finish. The sanding dust fills the pores. More work, and maybe not needed depending on your preference.
Here's a Martini International stock with no filling and more than ten thinned coats of Tru Oil -
When I bought that rifle I had no intention other than shooting, but the condition was so sorry I couldn't stand it and disassembled the gun as soon as I got home. I wish I had kept that rifle.
Stained with a Minwax oil based stain.