Therefore, to get the oil in all of the little grooves I had to really rag and push when I applied.
Next time don't be afraid to use a brush for application. The bristles of a brush are far, far better than a wad of cloth at getting finish (even a thin one like "Danish oil") into corners, around details and in fine grooves.
I'm 100% sure there's too much Danish Oil on the surface!
You have to be patient and dedicated when removing the excess of blended finishes like "Danish oil". It's not much different to when using straight oil, no matter how long it takes the surface shouldn't be wet when you're done1.
should I 1: follow the frequent suggestion to wipe down with thinner or meth or (worst case) acetone wipe?
I think this is the best course of action, and you want to do it ASAP.
The longer you wait the more insoluble the gummy finish is in thinners/mineral spirits, because the varnish component is oxidising and changing to its cured form (not soluble in mineral spirits). Note: in case you were thinking of using odorless mineral spirits the regular stuff has a noticeably stronger solvent action so although there's lots more smell it is to be preferred here.
Acetone can be required if the curing process has progressed far enough that the finish is only weakly dissolved by MS. If you do have to resort to using it for some or all of this, ensure good ventilation (air moving past you and out of the room is the ideal) or use a respirator with the appropriate cartridges fitted, and wear solvent-resistant gloves2.
Meths? Completely the wrong solvent for this.
Some how-to and tips:
You'll want to use plenty of paint thinner/mineral spirits and plenty of clean rags or paper towels doing this. If you try to skimp on solvent you won't dissolve enough of the finish residue. If you don't use enough changes of rags/paper towels what you end up doing is mostly spreading the finish around, rather than actually taking off what has been dissolved.
Don't be afraid to use a firm-bristled paintbrush or even an old toothbrush (spotlessly clean, or just a new one LOL) if necessary to help get excess oil out of the grooves.
The rags or paper towels used present the same fire hazard as the application rags do, which I presume the on-can instructions were very specific about. Arguably the easiest way to deal with them is to dry them flat on the floor somewhere, or on the ground outside if this is practical for you. After they've gone stiff they can be safely discarded in the trash.
2: Others say apply another coat of Danish oil and rag off hard again. (Intuitively, with the grooved wood, I fear this approach)
This can work (you'll see why down below where I strongly criticise "Danish oil" LOL) but really it's a sort of half measure. What you want to do is address the problem in the most direct manner, and that's to use solvent.
3: Others suggest using an alternative oil (like olive oil) with a hearty wipe-off.
Please do me a favour and never listen to anything else said by whatever source this was ^_^ You never want to apply olive oil to wood you're trying to finish like this as a host of problems can ensue.
4: Some say, hold firm, let it dry over time...slowly...
This might work. Heavy emphasis on might here. "Danish oil" is not one completely uniform product across all brands, and even with a single brand batch variation does occur, and they can even be reformulated (there's an infamous case of this from recent decades, involving the Watco one specifically IIRC!) so this advice is sort of sketchy.
So, it could dry enough that it stops being sticky and isn't soft enough to be a problem. But typically Danish oils are not hard enough to be applied thickly and left, they will remain permanently soft, even gummy, and the residue can be a dust magnet. There's no way to be sure beforehand which way this will go, and if you wait and it doesn't work you're left with an even more difficult problem to deal with.....
Recommendation for the future, ditch the "Danish oil"
I'll skip the boring historical perspective I originally wrote and just cut to the chase. Wiping varnish is a very viable alternative here, with some key advantages for you.
Purely as a finish in its own right varnish is superior, it makes the colour and figure in wood look virtually identical in most cases, and here's the kicker: it is easier to apply. You can use exactly the same basic method, but if you accidentally leave any excess in groves or a corner here or there it will reliably dry hard. As if that's not enough, you get way more product when you buy varnish3.
See more on wiping varnish in this previous Answer.
1 Good oiling guidelines stress that the surface should basically feel dry after the wiping off/buffing stage.
2 Acetone isn't that toxic on the skin but it will dry it out terribly if used in quantity over a longish time. Ever had to wash your hands in soap and water many times in succession and forgotten to moisturise? Like that.
3 Most commercial oil/varnish blends are heavily thinned in the factory (50% and sometimes quite a bit more is solvent!) so there's actually not much actual finish in the can o_O