I tried cleaning the top of the stone several times with Dawn dish liquid, after a few hours more oil oozes to the surface (visible in photo).
Keep going. Cleaning old stones is invariably a multi-step process, so you've already started on the road towards it being clean and just need to continue. There's more on cleaning from me in answer to this older Question, How can I identify sharpening stones?
As mentioned in the link above, one of the established ways of cleaning a stone is to soak in petrol (gasoline), kerosene or another solvent1. You can do this with the stone in the wood since organic solvents generally won't harm wood in any way, except sometimes removing a little colour2, but there's a good chance you'll lose the original finish with this method. Although initially very smelly fear not, the solvent smell will dissipate as it dries out and eventually the stone will have no odour to speak of.
It can take a heck of a lot of work to get all oil from a stone that has absorbed lots of it over decades of use, or was oil-filled in the factory in the case of synthetic stones. Expect the process to require numerous steps and to take a long time and just set to it. As there's no predicting how much oil needs to be removed and how much each step of cleaning will remove, it's one of those tasks that takes as long as it takes.
A friend cleaned out a Norton stone a year or so ago to see how it worked without the factory fill (many users report they cut much faster without and that's what he wanted). I believe it took a half-dozen soaks in boiling caustic water before it stopped releasing the original "oil filling"! Incidentally the factory fill is actually a grease, not a liquid oil, so there's a hint there — if one wants to fill a stone so that it doesn't continually absorb oil like a sponge don't soak it in oil but instead use something more like Vaseline3.
I'm open to removing the stone from the wood if the glue can be dissolved without harming the wood or stone, but would prefer to leave it as is. Not sure what kind of finish is on the wood, some kind of oil paint maybe.
It's likely you'd break the base if you tried to pry the stone out, and depending on the type of stone you risk breaking or chipping that too. Stones tend not to be glued in place BTW, historically they were set into plaster, although a user might use any glue or adhesive product he had to hand — casein, pitch, hide glue, epoxy, Bondo, you name it o_O
The safest way to get a stone free is to deliberately set out to lose the wooden case. It's of course up to each owner to decide for themselves which is more valuable, the stone or the case it's in.
You could try boiling water. This has been used to successfully release stones from wooden bases (works especially well if hide glue had been used) and the wood might dry later without harm, but old wood will often split after even brief soaking in hot water when it subsequently dries out. So although this seems like the gentlest option it still comes with a risk.
More info: the stone was severely dished when I got it, about 3/32" lower in middle than the sides.
I wondered at how flat it was for a stone of some vintage, you did a good job flattening it. Although a dished or 'swayed' stone can still be used (that's how they get so dished in the first place) stones are far more useful when as close to flat as possible.
I flattened on a wet cinder block, highly recommend that method, took about 10 minutes. Very fast, accurate, and cheap.
I've used that method too and was pleasantly surprised at how well it worked on some stones! Although hard naturals are a chore to abrade and one of the other established methods (using some form of commercial abrasive) may need to be adopted. A little use of abrasive is generally desirable to dress a stone's surface after flattening on something coarse4.
1 The solvent is contaminated with oil after this, so either needs to be reserved for cleaning other oily or greasy things (note: can be useful in the restoration of old tools) or A, disposed of in a suitable manner for your area, or B, left to evaporate until only the oil/grease remains and then that residue can be thrown out in the household refuse as it's no longer classed as toxic waste.
2 Oil-contaminated wood can actually be soaked in solvent for hours or days if necessary to try to get it clean, a technique sometimes used in the restoration of gun stocks.
3 The packaging of some older synthetic stones included instructions to soak in a pan of hot petroleum jelly. This was to temper cutting speed but it has the effect of keeping oil on the surface. I've found that just smearing a stone with Vaseline can do a lot to reduce its absorbency, although the soaking wouldn't be hard to arrange.
4 Stones can be made to cut a little faster or a lot slower depending on the surface texture, a process sometimes referred to as grading. On a single-grit stone (both natural and synthetic) this can be used to advantage to provide two faces with different properties.