...Y dando a un resorte...

[Quaeso]

¡Qué sudor triste cuándo se alejaban, en la lluvia, los pasos del cartero! Al fin una noche, me lo trajo. Era un breve aparato complicado, con lápiz, pluma, iniciales para lacre ... ¡qué sé yo! Y dando a un resorte, aparecía la estampilla, nuevecita, flamante. -Platero y Yo, Ch. XIV El Sello

Here I was hoping that you could offer a translation of the whole quote. But in particular I was having some issues with aparato, whether that was the package, or somehow the actual stamp mechanism; but I'm thinking it's the package containing the different stationary materials. And then secondly, with the phrase dando a un resorte, what that would mean.

[AngelicaDeAlquezar]

The speaker is sweating because he's excited when he sees the mailman coming to his house and when the man leaves without delivering the item he's expecting, he's disappointed. That's what "sudor triste" means.

This is not a formal translation, but here's what the paragraph could look like in English:
What a sad sweat it was when the mailman walked away in the rain!
He finally delivered it one night. It was a complex little device. It included a pencil, a pen, stamp letters for the sealing wax... I don't know what else! And triggering a spring mechanism, the bright brand new post stamp popped out.