Could you be more specific about how you've implemented it? Are you relying on the 2D physics engine in Godot, or are you programming your own? If it's the former, I'm not entirely sure how much I can help you there as things like wall jumping and other gravity defying features aren't exactly easy to program with built in physics (although hopefully, someone else who's done it will be along to explain how it works).
In the latter case, I have an implementation for that in my game (it's a bit of a mess because my game is kind of complicated, but you can try searching for wall_hanging where my game deals with it). I think I had to put a delay between detecting hanging to the wall and actually jumping after you press the jump button so that the program wouldn't aggressively try to clamp you to the wall when you try to leave it. My implementation is really basic though. It's not like the Megaman wall jump where you start sliding if you just hang to the wall.
