Learning Dutch is definitely a good idea, whether or not you will end up quite as happy with the results as you'd like. You owe it to yourself more than anything, especially once formal paperwork is involved. The idea that the Dutch will deal with English just fine is a bit of a misconception. Plenty do but you'll be encountering lots of old people (who grew up with a lot less of English in their lives and education); and even if they might be able to communicate in English in writing just fine, it is doubtful whether you'll understand most of them once they actually open their mouth.
I guess it's somewhere in between those two. On the one hand 4 months of sink-or-swim survival in The Hague with classes should probably cover the basics, you'll definitely won't need to fall back to point and grunt as often. On the other hand, Dutch is probably a bit of a counterintuitive language in many ways and it also relies heavily on idiom (which varies with dialect, and let's not venture across the border with Belgium just yet). Also, there's no hand holding with specific sentence structure, and the use of cases is only rudimentary. So if you compare it to a language like French or German, I'd rate Dutch as being a lot more difficult to learn well.
Fortunately, it seems the Dutch have reached some sort of consensus about that and sort of given up on it as well. So even if you mess up and butcher the grammar the Dutch are probably more likely to understand you (or be more forgiving about it) than the French or Germans. Sloppy grammar, poor style and "broken" sentences are, well, the rule rather than the exception much to the continued chagrin of language purists.
So I guess you should be alright if you put in effort, especially if you put in effort in pronunciation.
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