Is it worth it to keep watering the plants after they reach a harvestable state? does that increase the odds of getting more than 1 product, or should I just harvest when able?

well, yes, I noticed on the lemon and orange trees that if you water them constantly they will give fruit much faster. It is also tested on berry bushes, berries appear faster if watering is running constantly.

I'm more concerned with potatoes some other thingies.

Harvest as soon as you see the option.

I recommend saving beforehand as the plants yield a return of 1-3 chosen at random, so you can reload a save if you aren't satisfied with your harvest, for example planting 10 potatoes and getting 10 back. This can also save you from those rare times when you receive no yield from a plant at all; I have had that happen with sugar cane, so there is a non-zero chance of that happening with other plants.

i plant crops by the river or by a sprinkler and just forget about them. it might be luck, but the longer i forget, the more potatoes i get back?