Okay, so, according to my calculations... Yes. You are certifiably insane.

What if you just don't know the answer to one? Or if multiple answers fit, and you need letters to choose between them?

I usually go through in order until I get stuck and then switch directions to use the free letters.

For what its worth I do it the same as you. I feel accomplished when I get it all in one round but that only happens with the easy puzzles at the beginning of the week. I have a goal of completing a puzzle on just the first pass of accross clues without having to use any down clues.

That's what I do - try the across clues, then the down clues, then jump around to fill in anything I've missed

Sometimes I even try solving with just the across clues or just the down clues for an extra challenge (usually Monday or Tuesday puzzles)

It depends on the medium I'm using. If it's a book or newspaper, I'll jump around. If I'm using my phone it makes more sense to do all the across, then all down.

That's how I do it as well

I think that makes sense, at least for a first pass. I want all of the prompts stewing in my head, so I want to take them all in, and going in order is the most efficient way to do that. If I don’t get one I’ll just try it later after the others.