Making cold sesame noodles (NY Times recipe)

When you say "Sauce" I think what you might have there is Goma sauce. It's a premade sesame seed dressing--which is great on noodles, cold spinach, and salads.
It will some other things in there like sugar, mirin, an acid element like vinegar or lemon juice, and soy sauce. (sometimes dashi and miso).
So you might adjust the recipe if those elements are repeated. Personalty, the Kewpie brand is spot on for cold noodles right out of the bottle.

The Asian sesame pastes I've bought have been nothing but ground sesame seeds; the difference from Tahini is it's made from toasted, not raw, seeds - so its flavor is, well, more toasty and intense. Without seeing the label, I can't say for sure, but I'd imagine a product labeled 'sesame sauce' is just that - a prepared sauce: sesame paste and/or sesame oil, mixed with soy, ginger, garlic, etc. When I make sesame noodles, I use the pure paste and add the other ingredients fresh to taste - plus mirin/honey, and so on. If you're adding PB, I think you could go either way, just increasing the liquid ingredients (or whisking in a little water) as necessary to reach the consistency you want.

I'm not sure what actual sauce you bought so it might work as paste in some recipes but it depends. As far as the peanut butter, that's totally a matter of preference- crunchy can sometimes come off dry tasting so oil or the sauce may be needed to compensate for texture. Play around and see what you like best.