It's so frustrating when a new renovation is marred by something like this! Since warm water isn't working long-term, and you're concerned about discoloration, let's try a slightly different approach before resorting to soap.

Given it's a vinyl, grasscloth-like wallpaper, it should be fairly durable, but we still want to be gentle.

Here's what I'd suggest trying next:

1. Lukewarm Water and Gentle Blotting: Use a clean, non-linting cloth dampened with plain lukewarm water. Gently blot the areas with the paste residue. Avoid rubbing, as this can sometimes spread the paste or affect the texture. After blotting, immediately wipe the area with a clean, dry cloth. Since the water makes it disappear temporarily, this might lift some of the paste with repeated gentle blotting.

2. Very Diluted White Vinegar Solution (Test First!): White vinegar can sometimes help dissolve paste residue. Create a very dilute solution of white vinegar and water (e.g., 1 part white vinegar to 5 or 6 parts water). Before applying this to the visible areas, test it in a completely inconspicuous spot (like behind the toilet or low down near the baseboard) and wait a few minutes to ensure it doesn't affect the color or texture of your wallpaper. If it's safe, then gently dab the paste residue with a cloth dampened in the very diluted solution, followed by wiping with a clean, damp cloth (plain water) and then drying.

Regarding the spot where water doesn't even temporarily help: This is concerning and does suggest potential discoloration or a more stubborn substance. Try the gentle blotting with plain water on this spot first. If the diluted vinegar solution is safe based on your test, you could very carefully try it on this spot as well, using minimal solution and blotting.

Definitely avoid using dish soap directly at this stage unless the above methods fail and you've tested it in an inconspicuous area. Dish soap can sometimes leave a residue that's hard to remove from wallpaper.

How does this sound? Let me know if you try either of these methods and what the result is. We can troubleshoot further from there.