Oil on top of tahini does not automatically mean something has gone wrong. In many jars, the sesame solids settle and the oil rises. Taking only the top layer leaves the remaining paste increasingly dry.
Use a clean, sturdy spoon to bring the dense material up from the bottom. Go slowly at first; an enthusiastic stir can send oil over the rim while leaving the hard layer untouched. Once the texture is reasonably even, measure what you need.
Storage instructions differ, so follow the jar. If the flavor or smell seems rancid, do not try to cover it with lemon and garlic. Tahini contributes most of the character of a simple dressing, and a tired jar will announce itself. Remember that it is sesame, even when a recipe calls it only a creamy sauce.