Your trap is leaking at the fitting. See the gunk there?

thank you; what should I do about it? (I just tightened the white gear-shaped thing a bit, if that helps)

That's a start. Clean the crud and keep an eye on it please.

"be sure it has a proper, undamaged gasket in place."

There's probably not a gasket on that side.

Put a bowl instead of a paper towel under the trap to catch the water.

1. If it continues to leak, look at this picture.
2. Loosen nuts "C" and "A" and slide part "D" down.
3. Clean the crud from both surfaces where "C" and "D" meet.
4. Slide up "D" to meet "C". BE SURE THAT THE TWO PARTS LINE UP. Don't connect "D" to "C" at an angle.
5. Tighten "C" nut. With washer(s) in the correct position, tighten "A" nut.

so far the simple tightening seems to have worked, as I don't see anything in the bowl I put underneath (where the paper towel formerly was)

I think I would go ahead and remove the trap and clean it out. You don’t need any tools, just a bowl to catch the water in the trap. It might be that your trap is partially clogged and that back up water is what’s dripping out.

ok, I removed the trap and cleaned it out as best as I could; thanks for the advice everyone!

Here are some pics; am I missing a gasket?

No gasket is missing.