No, your damage is only halved once
If you are confused what a spell does, it is often a good idea to look at the full text. Absorb Elements (except the "At higher levels" part) says in the description section:
The spell captures some of the incoming energy, lessening its effect on you and storing it for your next melee attack. You have resistance to the triggering damage type until the start of your next turn. Also, the first time you hit with a melee attack on your next turn, the target takes an extra 1d6 damage of the triggering type, and the spell ends.
The first sentence that you quoted describes what the effect is overall, the next two sentences (bolded here) explain what that means in terms of games mechanics. There is not other game mechanical effect described for "lessening the effect", so the resistance is how the effect is lessened.
As per PHB, page 197, in case you have also another source of resistance to the same damage, the damage is only halved once:
Multiple instances of resistance or vulnerability that affect the same damage type count as only one instance. For example, if a creature has resistance to fire damage as well as resistance to all nonmagical damage, the damage of a nonmagical fire is reduced by half against the creature, not reduced by three-quarters.
Because of this, if you are already resistant to the kind of damage you absorb, the spell does not reduce the damage further. You do however get to store and re-deal the damage, because that sentence is a second, independent effect, and does not depend on the amount of damage the spell did (or did not) reduce.