[RPG] Is it possible to “charge up” a sorcerer/warlock by taking multiple short rests

Your spell slots recover based on the class feature that allows your spellcasting.

The language in the multiclassing section only mentions how you cast spells, not how you recover them. Note that recovery isn't mentioned at all in that paragraph. The spell slots are interchangeable when you're casting, but they recover based on their own mechanics. Unlike spell slots from multiple classes with Spellcasting, they don't merge into one pool; they remain separate except when you're actually casting out of them.

As a Warlock/Sorcerer, you have both the Pact Magic feature and the Spellcasting feature, which recover spell slots separately. Pact Magic slots recover on any rest, and Spellcasting slots recover on a long rest.

The big issue with this seems to rely on being able to stockpile spell slots through short rests during downtime. Unfortunately, in the Resting rules (page 67 of the basic rules), it states:

Adventurers can take short rests in the midst of an adventuring day and a long rest to end the day.

This leads me to believe that RAW, characters cannot take short rests during Downtime days and so the theoretical character has no chance to stockpile spell slots.