Try something like this for your "glass" shader. As Nathan mentioned, rasterized transparency will always be lacking compared to Raytracing, though and though EEVEE really doesn't like transparency through transparency, this shader attempts to strike a happy medium. Try it and see for yourself. Pay attention to the Blend Modes on the right, including the backface culling and show backface checkboxes (I just noticed Show backface is covered in my image, but it is un-checked):
However, glass is very much assisted (both in eevee and Cycles) by the use of an HDRI environment map. Switch to your World shading tab and add an Environment Texture (not Image Texture) with your HDRI. Blender's built in HDRI's are in the blender folder at .../2.93/datafiles/studiolights/world and many more can be found online for free. One of the most popular places is HDRI Haven (https://hdrihaven.com/hdris/).
If you want the effect from the lighting, but don't want the image in your background, you can separate them by using a Mix Shader, using the Is Camera Ray output of a Light Path node as the mix factor:
