Oh, oil finishes will do that (most of them), and you really need to take your time, start with a somewhat diluted oil (with mineral spirits), slop on a thick coat, wait a few minutes, wipe it off, and for the next day or so, continue to wipe off any that weeps out of the wood, then go to a somewhat less dilute oil, repeat, and so on until the last coat is undiluted oil, and should only need a few drops and should be rubbed in evenly. Buffing between coats (after weeping has stopped) also helps a lot. The appearancwe will also look blotchy for the first few coats and gradually even out, but when it's all done, you're left with a buttery soft finish that looks great, shows off the woodgrain fairly well, and just looks very classy.
It is a process, and it doesn't look very good until you have several coats on, so it does take time and patience -- at least with this method, which is how I learned it and practice, though I'm sure there are other methods and products too.
The wipe-on plys are the easiest and fastest, but nothing beats a great oil finish in my book.
Sorry for not explaining this earlier. Just google oil wood finishing and you should find lots of how-to, maybe even some closer to home that give local names for products and materials.
Also, no matter how well you sanded, I'd be very wary of trying a poly finish on woood that was previously oiled and I'd almost think it better to go back and do the oil again, or start with all new wood and a new base f you want to use poly.
I even have a spare base already made that needs a bit of touching up and a bit more poly if you want it.
Thanks,
Jim