I have used drying linseed oil before, with no other ingredients, and takes 1 full day to reach the state you have mentioned.

Try working with Walnut or Stand oil mixed with a tiny amount of OMS, and see how they react. I cannot recommend a Pre-made medium, I use a homemade recipe that contains 15% Amber varnish in Walnut oil.

Your medium may not be the issue.
I suspect that you may be using a thin glaze that may be absorbing
rapidly into the paint surface.Your medium/paint just doesn’t dry that fast,
but it can absorb instantly.
I would try couching with your medium first.

I’ll second Sid’s comment. In my experience, the ground has the greatest effect on drying time. The medium is less important. On a smooth, glassy surface, drying takes a good while even if your only “medium” is turps. On a very absorbent surface such as illo board, a thin layer of oil dries very fast.

Whatever you are painting on is absorbent.

My suggestion would be a clear isolation coat of — well, anything you choose. Liquin is as a good choice. Mind you, Alkyd is a funny thing: As an additive, it helps paint dry fast, but as an isolation layer, it can sometimes take a long, long time to dry completely — as much as a couple of weeks. I’m not saying that this will happen to you, but don’t be shocked if it happens.

Obviously, before you apply an isolation layer, you must be sure that any paint beneath it is absolutely dry.

If you are still searching for a slower-drying medium, I suggest walnut oil or safflower oil. Safflower is clearest of all and the slowest-drying of all. Make sure it’s the kind of safflower oil available at the art supply store, not the grocery! However, when it comes to walnut oil, you can go to the grocery store: I’ve had good luck with Spectrum Natural Walnut Oil, refined and with no additives.

How old is your linseed oil? What is the brand and type?

Test your oil on non-absorbent plastic. It must take several days to make a gel.
Use pure refined linseed oil for glazing. If your oil dries fast on painting surface, but stay wet on plastic, then check how much time you need to make well dried painting surface because glazing must be applied on DRIED oil paint and worst to use it on semidried oil painting surface.

Good advice.

The original post stated that the glaze medium was pure linseed oil. Is that really the case, or was there some solvent (turps, OMS, etc.) added to it?

Note that Liquin (and similar products) contains some solvent. That will interact with the underlying paint layer unless it is well-dried, not merely sort-of-dry. Pure oil is itself a solvent for other oil.

If the underlying paint layer is not well-dried, it will be softened by the glaze. And, presumably some of the active molecular fragments, generated as oil cures, will be mingled with the overlying glaze, causing it to dry faster. That will be especially true if the underlying not-yet-dry layer had certain siccatives.

Some pigments, such as the Umbers, contain Manganese compounds that act as a siccative. If your glaze contains any of these it will dry faster than it might otherwise dry, but yet I cannot imagine it drying as fast as you indicated.

This is a fast drying medium of linseed oil doped with some type of potent driers.

Ah. I had interpreted “regular drying linseed oil” to mean “ordinary linseed oil that should dry at its natural rate,” rather than “linseed oil with driers added.”

I assumed that it was [I]this Winsor Newton medium[/I] since he also mentioned using Liquin.

Corrected link is here.