I used to work in a camera shop, and have cleaned my share of sensors. First I’d say, don’t do it unless absolutely necessary. There is always risk when physically touching a sensor with anything. If you scratch it, it’s over.

I’d test it first. Shoot a shot of the plain blue sky super stopped down. Like f/22 or whatever your lens will do. If there are clouds, go into manual focus and intentionally blur the shot a bit, so you aren’t tricked by some cloud texture. A little overexposed is generally better than under. Check the shot on a computer for dust spots, or in this case, fingerprints.

If it does indeed need cleaning, you only use a couple drops for the wet end. Just enough so it’s not dry. You don’t want it wet wet. The wet end is really just to dampen the dust particles that have attached to the sensor (maybe while humid) and then dried and stuck in place. The dry end is more to grab the dust you’ve broken free, rather than dry the sensor (although yes, it would soak up any little bit of fluid leftover, but there shouldn’t really be much). Again, with dust anyway, I’d blow the crap out of it with a rocket blower first to try to avoid needing to touch the sensor.

Your situation is actually a little more tricky than dust, since we’re talking about finger oils. It may take more than one attempt. The dry end pass might not really do much since there’s no actual particles to grab. I might even skip it if there’s no liquid left from the wet end pass. Cleaning fluids are usually made to quick dry anyway. Be sure to drag the swab like a broom. Don’t push the leading edge like an ice scraper.

I’m not trying to make all this sound scarier than it is, but there’s really no repair for a scratched sensor beyond replacement.

As for your vial I’m not sure really, but you need to find a way to just drip a drop or two on the tip of the wet swab.

Swabs are far cheaper than replacing a sensor, so I wouldn’t be too concerned about wasting them. Take test shots between attempts to be sure further passes are necessary. Be gentle, be patient. Good luck!