Branching in MWI happens when quantum systems become entangled, not when time travellers arrive. I don't think MWI is going to save you here
That said, decoherence doesn't branch time in the sort of absolute manner you might be thinking of. "How many timelines exist" is a question similar to "how long is a coastline"; it depends on how closely you're looking. At the smallest scales, 'timelines' interfere and mingle in a way that distinct universes really shouldn't. It's only once decoherence has gone well out of control that they can be said to be truly separate, and even then the interference never goes to zero, only approximately zero.