Am I contractually obligated to follow that old handbook?
Yes. Consider this: your employer gave you a copy of the handbook saying, "here is a copy of the staff handbook." Because of this, you know that the document in question is the staff handbook of your company, regardless of the fact that it bears an old name, and that the policies it expresses are part of the terms of your employment contract.
If you were truly concerned that the document in question had no validity as the staff handbook, you perhaps should have expressed that concern by telling the person giving you the document that they appear to have given you the wrong document, because it bears a different name. By not doing this, you accept the terms in the document.
On the other hand, if you do do that, you'll just irritate the people who have to produce a rebranded staff handbook so they can give you a copy.