When the First Carrier says "yet our horse not packed" (Henry IV Part 1, II.i.3), the verb is easy to understand, because the sense of "load up" is close to the one we have today. But when the King says to Falstaff "Be packing" (Henry VI Part 1, IV.i.46), they are not talking about suitcases.
Here the meaning is "depart, be off".
In several contexts there is a completely unrelated set of senses, all to do with subterfuge. When Hamlet says, of Polonius, "This man shall set me packing" (Hamlet, III.iv.212), he means he could easily plot or scheme.
When Aaron tells Tamora's sons to find Muly, so that her black baby can be exchanged, he says "Go pack with him" (Titus Andronicus, IV.ii.154) - "make a secret arrangement".
And when Antony tells Eros that Cleopatra "has Packed cards with Caesar" (Antony and Cleopatra, IV.xiv.19) he means she has shuffled the cards in her favour. The card-sharp sense can still be heard, but the others fell out of use during the 17th century.