When you cast Dead Weight on a creature, is there a time frame to sacrifice the dead weight to another instant speed card effect before the dead weight goes into the graveyard but after it killed the creature it was cast on?
No. Once a player would get priority, if Dead Weight reduces the toughness of the creature to a value less than 0, or if the damage marked on the creature is equal to or greater than its (new) toughness, State Based Actions will cause the creature to go to the graveyard.
When the creature goes to the graveyard, Dead Weight will follow since it is no longer attached to an object or player.
No, there is no opportunity to sacrifice your Dead Weight as the creature it's attached to dies. An aura attached to a creature that dies will head to the graveyard effectively in the same moment as the creature dying (unless it's something special, like a Bestow creature from Theros).
Suppose the creature dies: Dead Weight reduced its toughness to 0 or less, or it brought the toughness low enough that the damage marked on that creature met or exceeded its lowered toughness. The next time anyone would get priority (i.e. usually immediately afterwards) we check state-based actions, and do all of them we can until there's none left to do. Here, that means we would:
- Move the creature to its owner's graveyard (for having 0 or less toughness [704.5f] or having enough damage to kill it [704.5g] among other possible causes)
- Move the Aura that is no longer attached to anything to its owner's graveyard [704.5n]
... then you get priority, but it's too late to do anything to sacrifice that enchantment.