« Quoi que ce soit » tends to be used in a negative sentence or in an if-clause, emphasising the notion of the smallest amount possible: "anything (+ at all!)" = "even the slightest bit!".

I would say something like:

1 : Je ne m’attendais pas à ce qu’il puisse accomplir quoi que ce soit par lui-même.

= "I didn’t expect he could achieve anything (at all) on his own."

2 : S’il lui arrivait quoi que ce soit, je t’en tiendrais responsable.

= "If anything (at all) happened to him, I’d hold you responsible."

Although « n’importe quoi » too can be translated into "anything", this is more like "anything (of whatever kind)".

« n’importe quoi » = "whatever you choose will suffice; it doesn’t matter"

« quoi que ce soit » = "even the slightest bit of something will affect the situation“.

So in the following instance, these two expressions are not interchangeable:

3 (o) : N’importe quoi fera l’affaire. = "Anything will do."

(x) : Quoi que ce soit fera l’affaire.

As for your example sentences:

avant de dire quoi que ce soit d’autre = "before saying even the slightest bit of anything else"

vs : avant de dire n’importe quoi d’autre = "before saying whatever else (of whatever kind)"