It depends wildly depending on the difficulty, my mindset and level of tiredness. This week I’m a bit busy so I went through the weekly Trails content without looking anything up and I certainly did not understand 100% of what was going on but I could understand enough to get the gist of (hopefully) everything.
My rule of thumb is always “if somebody asked you to summarize what just happened, could you do it?” If the answer is yes, then IMO you’re good. You don’t have to understand all the puns and secondary dialogues.
Learning to deal with the ambiguity and moving along is actually a useful skill to practice when learning a language, and probably even more so when learning Japanese specifically.
As I mentioned previously, a good thing with games like these is that the vast majority of the dialogue isn’t critical, and the stuff that’s important is usually repeated a lot (and you can always return and re-trigger conversations with important NPCs who will usually repeat the important parts).
And if it’s really too hard or frustrating, don’t hesitate looking up an English “let’s play” video on Youtube to see what is being said, and then try to match that with the Japanese you’re reading. It’s obviously much, much easier that way. I used that a lot when I played my first games.
But also, if you have the stamina for it, taking the time to look everything up and break sentences appart can be a very good exercise and you will learn a lot doing it, just don’t force yourself to do it all the time if it leads you to burnout.
So basically, there’s not really a wrong way to do it, just do what feels right.