Removing all ice counters from Thing in the Ice some other way will not cause it to transform. You’ll need to cast an instant or sorcery spell and cause its last ability to trigger.

An ability that triggers when a player casts a spell resolves before the spell that caused it to trigger. It resolves even if that spell is countered or otherwise leaves the stack without resolving.

When Thing in the Ice’s triggered ability transforms it, Awoken Horror’s ability will trigger and resolve before the spell that caused Thing in the Ice’s last ability to trigger.