What's a good approach to defense puzzles?

I've noticed that I struggle a lot with puzzles where your king is in check and your goal is to survive. Here's just the latest example (a really difficult one):

https://www.chess.com/puzzles/problem/1289167

Now the point isn't this example so much because I already checked the solution . I'm more wondering if anyone has a smart approach to this type of problem in general. I feel like they are overwhelming. 

Is there a good way to think about them?

When you look at the puzzle, you can see there are only three legal squares to move to. Two of them allow follow up checks. One does not.

So you start by trying to find a move that doesn't allow follow up checks? And then take that move as your prime candidate and start calculating from there? Do you then go back to calculate the other moves or are you satisfied if the move that doesn't allow follow up checks seems to work?

That's a nice idea btw