The ideal way to do this is to would be to hop into a time machine so you can begin again, and start by doing some colour testing on an offcut or a non-visible part of the board you're working on until you're happy with the result ;-) 1
Since that's not an option here are some alternatives.
Generally the ideal is to colour wood directly, which would involve taking the board back to bare wood as stains won't take properly to wood that has finish on it as you've already discovered. This is a fair amount of work of course, and you lose some material, but despite this I'd normally be recommending it except for the fact that you're working with red oak. Red oak has large open pores and may have absorbed the Black Walnut Danish Oil very deeply, especially from the end-grain surfaces.
So from where you are now I think your best option could be "gel stain", which despite what it's called is not stain but coloured varnish in gel form. I'm not really a fan of these personally, in part because of the stupid name, but they do what they're supposed to and are easy to work with — wipe on, wipe off. This can go directly on the wood after giving the oil sufficient time to dry, a couple/few days should do it2. You would need to protect this with a subsequent coat or two of varnish.
I don't think these are as viable but I'll mention in passing that spray lacquer and darker shades of shellac could also be used here. All three options will subdue the grain of the wood because it's being coated in a colour layer rather than the wood itself absorbing colour. This may or may not be desirable, but there's no alternative when you aren't working with bare wood that can be stained conventionally.
1 Seriously though, do finish tests. Especially when working with new finishing products (not only products you haven't used before, but those just bought as well since finishes have been known to vary batch to batch and occasionally the formulations are changed intentionally).
2 It may actually work fine after just a brief wait, but it's safer to be cautious and give the DO time to cure a bit.