QIf I put this follower into my EX area with its Last Words ability, does it keep its Fable counters?

ANo. Counters disappear when moving off the field.

When this follower moves to your cemetery with at least one Fable counter on it, its Last Words ability triggers. At the same time, the counters disappear, because they count as effects applied to the follower when it was on the field, and when a card moves zones, any applied effects from the previous zone don't carry over unless indicated otherwise. Then, you resolve this follower's Last Words ability, putting it into its owner's EX area.