Looks like my Godot Super Scaling add-on can somewhat fix this. The problem is with uneven scaling of pixel sprites, which will always be an issue in any engine (unless you scale by multiples of 2). However, with Godot Super Scaling, you can upscale the graphics, then downscale them back to native resolution and it looks much better. Still not perfect, but it doesn't look broken.

You can download the add-on from within the Godot Asset Library in the Godot Editor. Add the Super Scaling scene to your game scene, then place all of the game itself in a new Node2D node and link that in the Super Scaling property. Be sure to set Usage to 2D and the best settings are Scale Factor 2 and Smoothness 1. And check Enable On Play. Also make sure all your images have Filter turned Off. Hope that helps.