Wrong, even under the old rules. The Mate is a separate monster and you must Run Away from it as well as the original. We also removed the -1 penalty -- having to run twice is bad enough.
Well, you don't run away from the Mate specifically. You are -1 to run away from the pair, together. You either get away from the pair, or they both get you, in which case the original Bad Stuff applies. The Bad Stuff is not doubled in any way, except that it is 50% more likely.
The Mate's Bad Stuff is the same as the original monster's.